Pembrolizumab and Pralatrexate in Treating Patients With Relapsed or Refractory 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. =\<1 or at baseline) from adverse events due to agents administered more than 21 days earlier. * Prior chemotherapy, targeted small molecule therapy, or radiation therapy within 14 days prior to day 1 of therapy or who has not recovered (i.e. =\< 1 or at baseline) from adverse events due to a previously administered agent. \* Note: Subjects with =\< grade 2 neuropathy are an exception and may qualify for the study. * Antineoplastic biologic therapy or major surgery within 21 days of the first dose of trial medication. If subjects received major surgery more than 21 days ago, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Received live vaccine within 30 days prior to day 1 of protocol therapy. * Systemic steroid therapy or on any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. * Diagnosis of immunodeficiency. * Has a known additional malignancy that is progressing or requires active treatment. 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. * Congestive heart failure class III/IV according to the New York Heart Association (NYHA) Functional classification. * Known severe hypersensitivity reaction to pembrolizumab, pralatrexate, leucovorin or any excipients. * Active autoimmune disease that has required systemic treatment in the past 2 years (replacement therapies for hormone deficiencies are allowed). Hemolytic anemia associated with the lymphoma does not exclude a patient from the study. * Known history of human immunodeficiency virus (HIV) (HIV 1/2 antibodies). * Known active hepatitis B (e.g., hepatitis B virus surface antigen (HBsAg) reactive) or hepatitis C (e.g., hepatitis C virus ribonucleic acid \[HCV RNA\] \[qualitative\] is detected). * History of tuberculosis (TB) infection (Bacillus tuberculosis). * Active central nervous system (CNS) involvement by lymphoma, including parenchymal and/or lymphomatous meningitis. Subjects with prior CNS involvement by lymphoma must have a baseline magnetic resonance imaging (MRI) and lumbar puncture at screening that demonstrate no active lymphoma in the CNS. * Has a history of (non-infectious) pneumonitis that required steroids or current pneumonitis. * Active, uncontrolled infection requires systemic therapy. * Female: Pregnant or breastfeeding. * Expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. * Has known psychiatric or substance abuse disorder that would interfere with cooperation with the requirements of the trial. * 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. * Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures. * Perspective 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).