Inclusion Criteria:
* Histologically or cytologically confirmed untreated DEL and DHL diffuse large B-cell lymphoma (DLBCL) meeting the World Health Organization (WHO) criteria for DEL - MYC greater than 40% and BCL2 greater than 50% by immunohistochemistry, or high-grade B-cell lymphoma with MYC and BCL2 and/or BCL6 rearrangements (double-hit and/or triple-hit are included)
* Measurable disease by CT or PET/CT scan, with one or more sites of disease \>= 1.5 cm in longest dimension
* Age \>= 18 years at time of consent
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Life expectancy \>= 6 months
* Leukocytes \>= 2,500/uL
* Absolute neutrophil count \>= 1,000/uL
* Platelets \>= 100,000/uL
* Hemoglobin \>= 8 g/dL
* Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (however, patients with known Gilbert disease who have serum bilirubin level =\< 3 x ULN may be enrolled)
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x ULN (AST and/or ALT =\< 5 x ULN for patients with liver involvement)
* Alkaline phosphatase =\< 2.5 x ULN (=\< 5 x ULN for patients with documented liver involvement or bone metastases)
* Creatinine clearance \>= 30 mL/min by Cockcroft-Gault
* Activated partial thromboplastin time (aPTT) =\< 1.5 x ULN (This applies only to patients who do not receive therapeutic anticoagulation; patients receiving therapeutic anticoagulation, such as low-molecular-weight heparin or warfarin, should be on a stable dose)
* Transthoracic echocardiography (TTE) or multigated acquisition scan (MUGA) ejection fraction greater than 40%
* Women of child-bearing potential (WOCBP) must agree to use a highly effective method of contraception from the time of giving informed consent until at least 10 months after the last dose of study drug. Men with female partners who are of childbearing potential must agree to use a highly effective method of contraception from the time of giving informed consent until at least 7 months after the last dose of study drug
* Ability to understand and the willingness to sign a written informed consent document
* Human immunodeficiency virus (HIV) infected patients:
* No history of acquired immunodeficiency syndrome (AIDS)-defining conditions other than lymphoma or history of CD4+ T-cells below 200/mm\^3 prior to beginning combination anti-retroviral therapy (ART)
* Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial
* At time of study entry CD4+ T-cells must have recovered from prior lymphoma therapy to \>= 250/mm\^3
* At the time of study entry, the HIV viral load must be undetectable by standard laboratory assay
* During prior lymphoma therapy, patients must not have experienced documented infections attributed to the HIV positive (+) status
* No history of non-adherence to ART and willing to adhere to ART while on study
* Antiretroviral drugs with overlapping or similar toxicity profiles as study agents not allowed
Exclusion Criteria:
* Current/ prior use of:
* Lymphoma treatment, except for:
* 1 cycle of DA-EPOCH-R or rituximab, cyclophosphamide, doxorubicin (Adriamycin) vincristine (Oncovin) and prednisolone (R-CHOP)
* Radiotherapy \> 2 weeks of initiating study treatment
* Nitrosoureas or mitomycin C \> 6 weeks of initiating study treatment
* Steroid treatment for DLBCL or steroid monotherapy to stabilize disease while awaiting fluorescence in situ hybridization (FISH)
* Other cancer therapies (e.g., prostate, breast hormonal-based therapy) per the principal investigator's discretion
* Anthracycline greater than 50 mg/m\^2 (total lifetime) for a prior malignancy
* Complementary and alternative medications (CAM) within 1 week prior to initiating study treatment
* Treatment with any other investigational agent for any indication within 3 weeks prior to initiating study treatment
* Loncastuximab tesirine or rituximab with progression within 6 months of initiating study treatment
* Oral or intravenous (IV) antibiotics within 2 weeks prior to initiating study treatment. Patients receiving prophylactic antibiotics (e.g., for prevention of a urinary tract infection or chronic obstructive pulmonary disease) are eligible
* Live, attenuated influenza vaccine within 4 weeks prior to initiating study treatment
* Immunosuppressive medications (including, but not limited to, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor, such as anti-tumor necrosis factor \[TNF\] agents) within 14 days prior to initiating study treatment. The following are exceptions to this criterion:
* Steroids
* Bisphosphonate therapy for symptomatic hypercalcemia or for other reasons (e.g., bone metastasis or osteoporosis)
* Known uncontrolled central nervous system (CNS) involvement by lymphoma, including leptomeningeal involvement
* History of hypersensitivity to anti-CD19 antibodies, loncastuximab tesirine, or any agents used in DA-EPOCH-R
* History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to other agents used in study
* Clinically significant third space fluid accumulation (i.e., ascites requiring drainage or pleural effusion that is either requiring drainage or associated with shortness of breath)
* Breastfeeding or pregnancy
* Clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; or inherited liver disease
* Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HbsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible
* Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA)
* Documented eczema, psoriasis, or lichen simplex chronicus of vitiligo with dermatologic manifestations (e.g., patients with psoriatic arthritis would be excluded), unless the following apply:
* Affected skin covers less than 10% of body surface area (BSA)
* Disease is well controlled at baseline and only requires low potency topical steroids (e.g., hydrocortisone 2.5%, hydrocortisone butyrate 0.1%, fluocinolone 0.01%, desonide 0.05%, alclometasone dipropionate 0.05%)
* No acute exacerbations of underlying condition within the last 12 months (not requiring psoralen plus ultraviolet A radiation \[PUVA\], methotrexate, retinoids, biologic agents, oral calcineurin inhibitors; high potency or oral steroids)
* Known active tuberculosis (TB)
* Severe infections within 4 weeks prior to initiating study treatment, including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia
* Major surgical procedure within 28 days prior to initiating study treatment or anticipation of need for a major surgical procedure during the course of the study
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
