Nivolumab & Brentuximab Vedotin Consolidation After Autologous SCT in Patients With High-Risk Classical Hodgkin Lymphoma

Inclusion Criteria: * Documented informed consent * Agreement to allow the use of archival tissue from pre-ASCT tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Histologically confirmed diagnosis of classical Hodgkin lymphoma (excluding nodular lymphocyte predominant Hodgkin lymphoma) according to the World Health Organization (WHO) classification, with hematopathology review at the participating institution * Have high-risk relapsed or refractory Hodgkin lymphoma (HL), defined as at least one of the following: * Primary refractory disease to front-line therapy * Relapse within 1 year of completing front-line therapy * Extranodal involvement at the time of pre-ASCT relapse * B symptoms at pre-ASCT relapse * More than one type of pre-ASCT salvage therapy required * Planning to receive or have received autologous stem cell transplantation (ACST) per institutional standards as part of standard of care * Pre-ASCT participants may consent but will not be eligible to begin treatment until after ASCT, and will have to fulfill all inclusion and exclusion criteria before starting protocol * All participants must initiate day 1 of protocol therapy within 30-60 days post stem cell reinfusion; study PI can grant exception for a patient to start as late as 75 days post stem cell reinfusion with a reasonable justification for a delay (e.g. recovery from post -ASCT toxicity) and this will not be a protocol deviation, nor require an exception to be filled * Recovery from ASCT toxicity as defined as outpatient status, able to drink, eat normally, and do not need intravenous hydration prior to day 1 of therapy * Achieved at least stable disease to salvage treatment determined by positron emission tomography (PET)/computed tomography (CT) using 2014 Lugano Classification prior to ASCT * Brentuximab vedotin naive OR had at least stable disease by Lugano Classification to prior brentuximab vedotin treatment * Absolute neutrophil count (ANC) \>= 1000/mm\^3 * Platelets \>= 50,000/mm\^3 * Hemoglobin \>= 8 g/dL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) or 3 x ULN for Gilbert's disease * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN * Creatinine clearance \>= 40 mL/min per 24 hour urine collection or the Cockcroft-Gault formula * Calculated per institutional standard * Forced expiratory volume in one second (FEV1) and carbon monoxide diffusion capacity (DLCO) (adjusted for hemoglobin \[Hb\]) \>= 50% adjusted * Women of childbearing potential (WOCBP) only: Negative urine or serum pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Woman of childbearing potential (WOCBP): use two effective methods of contraception (hormonal or barrier method) or be surgically sterile, or abstain from heterosexual activity for the course of the study through 7 months post last dose of nivolumab * WOCBP defined as not being surgically sterilized or have not been free from menses for \> 1 year * Male: use two effective methods of contraception (barrier method) or abstain from heterosexual activity with the first dose of study therapy through 7 months post last dose of nivolumab Exclusion Criteria: * Post-ASCT anti-lymphoma or investigational therapy; immediate post-ASCT consolidative radiation therapy is allowed as long as it occurs prior to initiation of study therapy; baseline imaging and pulmonary function tests (PFTs) must be performed after completion of radiation * Previous allogeneic transplant * Total carmustine (BCNU) dose of \> 600 mg/m\^2 with prior treatments including transplant conditioning regimen * Live vaccine within 30 days prior to day 1 of protocol therapy (e.g. measles, mumps, rubella, varicella, yellow fever, rabies, Bacillus Calmette-Guerin \[BCG\], oral polio vaccine, and oral typhoid) * Refractory to prior brentuximab vedotin (i.e. progression while on treatment) * Refractory to prior anti-PD-1/PD-L1 agent * History of prior \>= grade 3 hypersensitivity to either brentuximab vedotin or nivolumab * History of another primary malignancy that has not been in remission for at least 3 years; exceptions include: * Basal cell carcinoma of the skin or * Squamous cell carcinoma of the skin that has undergone potentially curative therapy or * In situ cervical cancer * Known active central nervous system (CNS) involvement by lymphoma, including parenchymal and/or lymphomatous meningitis * History of progressive multifocal leukoencephalopathy (PML) * Grade \>= 2 peripheral neuropathy at the present time * Prior diagnosis of inherited or acquired immunodeficiency * Condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone or equivalent) or other immunosuppressive medications within 14 days of study drug administration; exceptions are: * Inhaled or topical steroids and * Adrenal replacement doses \> 10 mg daily prednisone equivalents in the absence of active autoimmune disease * Uncontrolled illness including ongoing or active infection * History of or active pneumonitis or interstitial lung disease: * For history of pneumonitis to be an exclusion, patient had to have required supplemental oxygen or corticosteroid treatment; radiographic changes alone are not an exclusion * An active, known or suspected autoimmune disease; the following are exceptions: * Vitiligo * Hemolytic anemia associated with the lymphoma (history of or at the present time) * Type I diabetes mellitus * Residual hypothyroidism due to autoimmune condition only requiring hormone replacement * Psoriasis not requiring systemic treatment, or * Conditions not expected to recur in the absence of an external trigger * Active or known history (standard pre-ASCT assessments) of: * Hepatitis B or C infection * Human immunodeficiency virus (HIV) * Acquired immunodeficiency syndrome (AIDS) * Women who are pregnant or lactating * History of a cerebral vascular event (stroke or transient ischemic attack), unstable angina, myocardial infarction, or cardiac symptoms consistent with New York Heart Association Class III-IV within 6 months prior to day 1 of protocol therapy * History or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)