• Untreated aggressive B-cell large-B cell lymphoma (non-Hodgkin lymphoma) with adverse features that may predict sub-optimal response to rituximab-cyclophosphamide, doxorubicin (hydroxydaunorubicin), vincristine (Oncovin), prednisone (R-CHOP) and in the opinion of the investigator would be treated with DA-EPOCH-R as standard of care. Subjects must be planned to receive full course (6 cycles) chemoimmunotherapy as per clinical standard of care. Composite lymphomas are not excluded provided that the subject has not receive prior systemic therapy for the indolent component and would receive DA-EPOCH-R as the standard of care regimen for the aggressive component. Eligible histologies based on 2016 World Health Organization (WHO) classification include:

‣ High grade B-cell lymphoma with MYC and BCL2 and/or BCL6 translocations

⁃ High grade B-cell lymphoma, not otherwise specified (NOS)

⁃ Diffuse large B-cell lymphoma (DLBCL) NOS

⁃ Primary mediastinal B-cell lymphoma

⁃ T-cell/histiocyte-rich large-B-cell lymphoma

⁃ Epstein-Barr virus (EBV) + DLBCL, NOS

⁃ ALK+ large B-cell lymphoma (must be CD20+)

⁃ B-cell lymphoma, unclassifiable, with features intermediate between DLBCL and classical Hodgkin lymphoma

• Be willing and able to provide written informed consent for the trial

• Be \>= 18 years of age on day of signing informed consent

• Have measurable disease, including at least 1 nodal site measuring 1.5 cm or 1 extranodal site measuring 1.0 cm in longest dimension on computed tomography (CT) or fluorodeoxyglucose-positron emission tomography (FDG-PET)

• Have a performance status of 0-2 on the Eastern Cooperative Oncology Group (ECOG) performance scale (PS)

• Left ventricular ejection fraction (LVEF) \>= 50% on cardiac multiple-gated acquisition (MUGA) scan or cardiac echocardiogram (ECHO)

• Absolute neutrophil count (ANC) \>= 1,000/uL except in cases of marrow infiltration by lymphoma

• Platelets \>= 75,000 / mcL except in cases of marrow infiltration by lymphoma or hypersplenism

• Hemoglobin \>= 8 g/dL except in cases of marrow infiltration by lymphoma without red blood cell (RBC) transfusion within 14 days of first treatment

• Measured or calculated creatinine clearance (Glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) \>= 40 mL/min.

• \* Creatinine clearance should be calculated per institutional standard

• Serum total bilirubin =\< 1.5 X upper limit of normal (ULN) (Patients with documented Gilbert disease may be enrolled if total bilirubin =\< 3.0 x ULN)

• Direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 ULN

• Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN OR =\< 5 X ULN for subjects with liver involvement

• International Normalized Ratio (INR) or Prothrombin Time (PT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants

• Activated partial thromboplastin time (aPTT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants, or subject is shown to have an antiphospholipid antibody on workup

• For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of =\< 1% per year during the treatment period and for at least 12 months after the last dose of EPCH-R, 9 months after the last dose of polatuzumab, 2 months after the last dose of glofitamab (Arm B participants), or 3 months after the last dose of tocilizumab (if applicable), whichever is longer. Women must refrain from donating eggs during this same period. A woman is considered to be of childbearing potential if she is post-menarcheal, has not reached a postmenopausal state (=\<12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus). The definition of childbearing potential may be adapted for alignment with local guidelines or requirements. Examples of contraceptive methods with a failure rate of =\<1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post-ovulation methods) and withdrawal are not acceptable methods of contraception

• For women of childbearing potential, a negative serum pregnancy test result during screening period. Women who are considered not to be of childbearing potential are not required to have a pregnancy test

• For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm, as defined below: With female partners of childbearing potential or pregnant female partners, men must remain abstinent or use a condom during the treatment period and for at least 6 months after the last dose of EPCH-R or polatuzumab, 2 months after the last dose of glofitamab (Arm B participants), or 2 months after the last dose of tocilizumab (if applicable), whichever is longer. Men must refrain from donating sperm during this same period. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of preventing drug exposure. Male patients considering preservation of fertility should bank sperm before study treatment

• ARM B ONLY: Negative SARS-CoV-2 antigen or PCR test within 7 days prior to enrollment