Inclusion Criteria:
* Patients must be \> 12 years and =\< 39 years of age at the time of study enrollment
* Patients must have had histologic verification of osteosarcoma at original diagnosis or relapse
* Patients with diagnoses of osteosarcoma and confirmed HER2 expression of \> 10% of osteosarcoma cells are eligible for the intervention
* Note: There is a mandatory tissue submission for HER2 staining during the Step 0 Eligibility Screening process. Metastatic tissue, when possible from the most recent relapse, is strongly preferred for HER2 staining over archival tissue from primary resection or diagnostic biopsy. The evaluation period for HER2 staining to determine eligibility for therapy will be less than 4 weeks from screening enrollment
* Patients must have measurable disease according to Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. Patients with clinically inactive brain metastases may be included in the study. Patients with treated brain metastases that are no longer symptomatic and who require no treatment with corticosteroids or anticonvulsants may be included in the study if they have recovered from the acute toxic effect of radiotherapy. Lastly, patient must have unresectable lesions or lesions with no intention to surgically remove in the 6 months following enrollment
* Patient's current disease state must be one for which they have received at least standard initial therapy, defined as systemic therapy combined with either radiation or surgery for local control of the primary tumor at diagnosis. Prior therapy after relapse is not required
* Patients must have a performance status corresponding to Eastern Cooperative Oncology Group (ECOG) scores of 0 or 1. Use Karnofsky for patients \> 16 years of age and Lansky for patients =\< 16 years of age. Patients who are non-ambulatory as a result of prior surgical treatment for osteosarcoma should be considered ambulatory for the purposes of assessing performance status
* Patients must have recovered from the acute toxic effects of all prior anti-cancer therapy and must meet the following minimum duration from prior anti-cancer directed therapy prior to enrollment. If after the required timeframe, the numerical eligibility criteria are met, e.g., blood count criteria, the patient is considered to have recovered adequately
* Cytotoxic chemotherapy or other anti-cancer agents known to be myelosuppressive: For agents not listed, the duration of this interval must be discussed with the study chair and the study-assigned Research Coordinator prior to enrollment
* \>= 21 days after the last dose of myelosuppressive chemotherapy (42 days if prior nitrosourea)
* Anti-cancer agents not known to be myelosuppressive (e.g., not associated with reduced platelet or absolute neutrophil count \[ANC\] counts): \>= 7 days after the last dose of agent
* Antibodies: \>= 4 weeks (28 days) must have elapsed from infusion of last dose of antibody, and toxicity related to prior antibody therapy must be recovered to grade =\< 1
* Corticosteroids
* Hematopoietic growth factors: \>= 14 days after the last dose of a long-acting growth factor (e.g. pegfilgrastim) or 7 days for short-acting growth factor. For agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur
* Interleukins, interferons and cytokines (other than hematopoietic growth factors): \>= 21 days after the completion of interleukins, interferon or cytokines (other than hematopoietic growth factors)
* Stem cell Infusions (with or without total body irradiation \[TBI\]):
* Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including donor lymphocyte infusion (DLI) or boost infusion: \>= 84 days after infusion and no evidence of graft versus host disease (GVHD)
* Autologous stem cell infusion including boost infusion: \>= 30 days
* Vellular therapy: \>= 30 days after the completion of any type of cellular therapy (e.g., modified T cells, natural killer \[NK\] cells, dendritic cells, etc.)
* Radiation therapy (XRT)/external beam irradiation including protons: \>= 4 weeks (28 days) including palliative radiation therapy to the chest. \>= 14 days after palliative local XRT to areas other than the chest or for whole brain radiotherapy
* Radiopharmaceutical therapy (e.g., radiolabeled antibody, samarium): \>= 42 days after systemically administered radiopharmaceutical therapy
* Patients must not have received prior HER2 therapies including antibody drug conjugates (e.g. TDM-1 or DS-8201a), HER2 directed cellular therapies, HER2 receptor therapy (e.g. trastuzumab, pertuzumab) or small molecule antagonists of HER2 (e.g lapatinib or neratinib)
* Patients must be at least 7 days from the date of last surgery
* Peripheral absolute neutrophil count (ANC) \>= 1000/mm\^3, (granulocyte colony-stimulating factor \[G-CSF\] administration is not allowed within 1 week prior to Step 1 screening assessment) (for patients with solid tumors without known bone marrow involvement)
* Platelet count \>= 100,000/mm\^3 (transfusion independent, defined as not receiving platelet transfusions for at least 7 days prior to enrollment) (for patients with solid tumors without known bone marrow involvement)
* Hemoglobin \>= 8.0 g/dL at baseline (Red Blood Cell transfusion is not allowed within 1 week prior to screening assessment) (for patients with solid tumors without known bone marrow involvement)
* Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 or a serum creatinine based on age/gender as follows:
* Age: 12 to \< 13 years; Maximum serum creatinine (mg/dL): 1.2 (male); 1.2 (female)
* Age: 13 to \< 16 years; Maximum serum creatinine (mg/dL): 1.5 (male); 1.4 (female)
* Age: \>= 16 years; Maximum serum creatinine (mg/dL): 1.7 (male); 1.4 (female)
* For participants less than 18 years of age that screen fail only based on creatinine, a 24 hour urine collection may be used instead to confirm eligibility. A calculated GFR \> 60 mL/min/1.73 m\^2 using a 24 hour collection will meet criteria for inclusion on this trial
* Bilirubin (sum of conjugated + unconjugated or total) =\< 1.5 x upper limit of normal (ULN) for age. For patients with documented Gilbert's syndrome (unconjugated hyperbilirubinemia) bilirubin must be \< 3 x ULN for age (patients with solid tumors)
* Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) and serum glutamic-oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) =\< 3 x ULN. (if liver metastases present =\< 5 x ULN). For the purpose of this study, the ULN for ALT is 45 U/L regardless of baseline and the ULN for AST is 50 U/L regardless of baseline (patients with solid tumors)
* Serum albumin \>= 2.5 g/dL (patients with solid tumors)
* International normalized ratio (INR)/prothrombin time (PT) and either partial thromboplastin or activated partial thromboplastin time (aPTT) =\< 1.5 x ULN (patients with solid tumors)
* Shortening fraction of \>= 27% by echocardiogram, or ejection fraction of \>= 50% by either an echocardiogram (ECHO) or multigated acquisition (MUGA) scan within 28 days before Step 1 enrollment
* Corrected QT interval (QTc) prolongation to \< 470 ms (females) or \< 450 ms (males) based on average triplicate 12-lead electrocardiogram (ECG)
* Pulse oximetry \> 93% on room air
* Patients with seizure disorder may be enrolled if on anticonvulsants and well controlled as evidenced by no increase in seizure frequency in the prior 7 days
* Nervous system disorders (Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\]5) resulting from prior chemotherapy, surgery, and/or radiation must be =\< grade 2, with the exception of decreased tendon reflex (DTR). Any grade of DTR is eligible
* All patients and/or their parents or legally authorized representatives must sign a written informed consent. Assent, when appropriate, will be obtained according to institutional guidelines
Exclusion Criteria:
* Pregnant, planning to become pregnant, or breast-feeding women will not be entered on this study because there is yet no available information regarding human fetal or teratogenic toxicities. Pregnancy tests must be obtained in girls who are post-menarchal. Males or females of reproductive potential may not participate unless they have agreed to use two effective methods of birth control, including a medically accepted barrier or contraceptive method (e.g., male or female condom) for the duration of the study and upon completion of the study and for at least 7 months for females and 4 months for males after the last dose of study drug. Abstinence is an acceptable method of birth control
* Methods considered as highly effective methods of contraception include:
* Combined (estrogen and progestogen containing) hormonal contraception associated with inhibition of ovulation:
* Oral
* Intravaginal
* Transdermal
* Progestogen-only hormonal contraception associated with inhibition of ovulation:
* Oral
* Injectable
* Implantable
* Intrauterine device (IUD)
* Intrauterine hormone-releasing system (IUS)
* Bilateral tubal occlusion
* Vasectomized partner
* Complete sexual abstinence defined as refraining from heterosexual intercourse. Periodic abstinence (calendar, symptothermal, post-ovulation methods) is not an acceptable method of contraception
* Non-child-bearing potential defined as pre-menopausal females with a documented tubal ligation or hysterectomy; or postmenopausal defined as 12 months of spontaneous amenorrhea (in questionable cases, a blood sample with simultaneous follicle-stimulating hormone \[FSH\] \> 40 mIU/mL and estradiol \< 40 pg/mL \[\< 147 pmol/L\] is confirmatory). Females on hormone replacement therapy (HRT) and whose menopausal status is in doubt will be required to use one of the contraception methods outlined for women of child-bearing potential if they wish to continue their HRT during the study. Otherwise, they must discontinue HRT to allow confirmation of post-menopausal status prior to study enrollment. For most forms of HRT, at least 2-4 weeks will elapse between the cessation of therapy and the blood draw; this interval depends on the type and dosage of HRT. Following confirmation of their post-menopausal status, they can resume use of HRT during the study without use of a contraceptive method
* Male subjects must not freeze or donate sperm starting at Screening and throughout the study period, and at least 4 months after the final study drug administration. Preservation of sperm should be considered prior to enrolment in this study
* Female subjects must not donate, or retrieve for their own use, ova from the time of Screening and throughout the study treatment period, and for at least 7 months after the final study drug administration
* Patients receiving corticosteroids who have not been on a stable or decreasing dose of corticosteroid for at least 7 days prior to enrollment are not eligible. If used to modify immune adverse events related to prior therapy, \>= 14 days must have elapsed since last dose of corticosteroid
* Patients who are currently receiving another investigational drug are not eligible
* Patients who are currently receiving other anti-cancer agents are not eligible
* Patients who are receiving cyclosporine, tacrolimus or other agents to prevent graft-versus-host disease post bone marrow transplant are not eligible for this trial
* Patients who are receiving chloroquine or hydroxychloroquine within 14 days are not eligible for this trial
* Patients who have received a prior solid organ transplantation are not eligible
* Patients with a medical history of myocardial infarction within 180 days before enrollment, symptomatic congestive heart failure (CHF) (New York Heart Association Class II to IV) or troponin levels consistent with myocardial infarction as defined according to the manufacturer 28 days prior to enrollment are not eligible
* Patients who have a pleural effusion, ascites or pericardial effusion that requires drainage, peritoneal shunt, or Cell-free and Concentrated Ascites Reinfusion Therapy (CART) are not eligible
* Patients who have spinal cord compression or clinically active central nervous system metastases, defined as untreated and symptomatic, or requiring therapy with corticosteroids or anticonvulsants to control associated symptoms are not eligible
* Patients with a known history of severe hypersensitivity to DS-8201a or any excipient contained in the DS-8201a drug formulation are not eligible
* Patients who have an uncontrolled infection or non-healing surgical site are not eligible
* Patients with a known history of substance abuse or any other clinically significant medical conditions (i.e. psychological conditions) that may, in the opinion of the investigator, interfere with the patient's participation in the clinical study or evaluation of the clinical study results are not eligible
* Patients who have pulmonary compromise, ex hypoxia, resulting from intercurrent pulmonary illnesses including, but not limited to, any underlying pulmonary disorder (i.e. pulmonary emboli within three months of the study enrollment, severe asthma, severe chronic obstructive pulmonary disease (COPD), restrictive lung disease, pleural effusion etc.), or prior pneumonectomy are not eligible
* Patients who have a history of (non-infectious) ILD (interstitial lung disease)/pneumonitis that required steroids, has current ILD/pneumonitis, or where suspected ILD/pneumonitis cannot be ruled out by imaging at screening are not eligible
* Patients who have a pleural effusion, ascites or pericardial effusion that requires drainage, peritoneal shunt, or cell-free and concentrated ascites reinfusion therapy (CART) are not eligible. (Drainage and CART are not allowed within 2 weeks prior to screening assessment)
* Patients who, in the opinion of the investigator, may not be able to comply with the safety monitoring requirements of the study are not eligible
