Pembrolizumab and Pralatrexate in Treating Patients With Relapsed or Refractory Peripheral T-Cell Lymphomas

This phase I/II trial studies the side effects and best dose of pralatrexate when given together with pembrolizumab and how well they work in treating patients with peripheral T-cell lymphomas that has come back after a period of improvement or has not responded to treatment. Pralatrexate may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Giving pembrolizumab and pralatrexate may work better in treating patients with peripheral T-cell lymphomas.

Inclusion Criteria: * Documented willingness and ability to sign an informed consent of the participant and/or legally authorized representative. * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. * Patients must have a histologically confirmed diagnosis of mature peripheral T-cell or natural killer (NK)-cell lymphoma according to the World Health Organization (WHO) classification, with hematopathology review at the participating institution. Eligible histologies are: * Peripheral T-cell lymphoma, not otherwise specified * Anaplastic large cell lymphoma, ALK-negative * Anaplastic large cell lymphoma, ALK-positive * Angioimmunoblastic T-cell lymphoma * Nodal peripheral T-cell lymphoma with TFH phenotype * Follicular T-cell lymphoma * Indolent T-cell lymphoproliferative disorder of the gastrointestinal (GI) tract * Extranodal NK-/T-cell lymphoma * Enteropathy-associated T cell lymphoma * Monomorphic epitheliotropic intestinal T-cell lymphoma * Hepatosplenic T-cell lymphoma * Subcutaneous panniculitis-like T-cell lymphoma * Transformed mycosis fungoides * Patients must have failed at least one prior regimen, including: * Recurrence of disease after a documented complete response (CR). * Progression of disease after a partial response (PR) to the prior regimen. * Partial response, stable disease (SD) or progressive disease (PD) at the completion of the prior treatment regimen. If a patient has PR to prior regimen without PD, there must be biopsy-proven\*\* residual disease that is measurable * Exception can be granted by the principal investigator (PI) if a biopsy is not feasible and/or safe * Patient must have measurable disease by computerized tomography (CT) or positron emission tomography (PET) scan, with one or more sites of disease \>= 1.5cm in longest dimension. * Be willing to provide tissue from a fresh core or excisional biopsy of a tumor lesion prior to starting study therapy or from archival tissue of a biopsy that was performed after the most recent systemic therapy. Exception can be granted by the PI if a biopsy is not feasible and/or safe * Patients must have received one dose of vitamin B12 (1 mg intramuscularly \[IM\]) within 10 weeks prior to first dose of pralatrexate, and must have begun folic acid supplementation (1 mg orally, once daily) within 10 days of first dose of pralatrexate. Note: If folic acid was not started but methylmalonic acid (MMA) and homocysteine (HCY) levels were checked and are in normal range at screening, the investigator can decide to start study therapy immediately. Vitamin B12 and folic acid supplementation is standard of care for pralatrexate therapy. * Absolute neutrophil count (ANC) \>= 1,000/mm\^3. In Phase 2 portion of study, ANC \< 1000/mm\^3 but \>= 500/mm\^3 is allowable if patients have demonstrated bone marrow involvement by lymphoma (within 14 days prior to day 1 of protocol therapy). * Note: Growth factor is not permitted within 7 days of ANC assessment unless cytopenia is secondary to disease involvement. * Platelets \>= 75,000/mm\^3. In Phase 2 portion of study, Platelets \< 75,000/mm\^3 but \>= 25,000/mm\^3 is allowable if patients have demonstrated bone marrow involvement by lymphoma (within 14 days prior to day 1 of protocol therapy) * Note: Platelet transfusions are not permitted within 7 days of platelet assessment unless cytopenia is secondary to disease involvement. * Total serum bilirubin =\< 1.5 X upper limit of norma (ULN) or =\< 3X ULN for Gilbert's disease. Direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 ULN (within 14 days prior to day 1 of protocol therapy). * Aspartate aminotransferase (AST) =\< 2.5 x ULN (within 14 days prior to day 1 of protocol therapy) * Alanine aminotransferase (ALT) =\< 2.5 x ULN (within 14 days prior to day 1 of protocol therapy) * Creatinine clearance of \>= 60 mL/min per the Cockcroft-Gault formula (within 14 days prior to day 1 of protocol therapy). If creatinine clearance (CrCl) is \>= 60 mL/min as measured by 24 hour urine collection, this will be allowable. * If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) =\< 1.5 x ULN (within 14 days prior to day 1 of protocol therapy). * If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants. * If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) =\< 1.5 x ULN (within 14 days prior to day 1 of protocol therapy) * If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants. * Female of childbearing potential: negative urine or serum pregnancy test within 72 hours prior to receiving the first dose of study medication (within 14 days prior to day 1 of protocol therapy). 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 120 days post last dose of pembrolizumab. 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 120 days post last dose of pembrolizumab. Exclusion Criteria: * Patients with adult T-cell leukemia/lymphoma * Prior allogeneic hematopoietic stem cell transplantation within the last 5 years. * Prior autologous hematopoietic stem cell transplant within the last 60 days. * Patients who received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent without having had evidence of objective response. * Patients who received prior therapy with pralatrexate without having had evidence of objective response. * Investigational agent or anti-cancer monoclonal antibody (mAb) within 21 days prior to day 1 of therapy or who has not recovered (i.e. =\