huJCAR014 CAR-T Cells in Treating Adult Patients With Relapsed or Refractory B-Cell Non-Hodgkin Lymphoma or Acute Lymphoblastic Leukemia

Inclusion Criteria: * Male or female \>= 18 years of age at the time of screening consent INCLUSION CRITERIA FOR SCREENING * Diagnosis of R/R B-cell NHL or ALL as defined below: * Relapsed or refractory B-cell NHL meeting all of the following criteria: * Diffuse large B-cell lymphoma (DLBCL), not otherwise specified (NOS); high grade B cell lymphoma with MYC and BCL2 and/or BCL6 rearrangements; LBCL transformed from any indolent histology; or primary mediastinal B-cell lymphoma (PMBCL) * Prior treatment with an anthracycline and rituximab or another CD20-targeted agent (unless the disease is CD20-negative); transformed DLBCL (tDLBCL) must have failed treatment for DLBCL * At least one of the following: * Refractory disease after frontline chemo-immunotherapy * Not eligible for autologous hematopoietic stem cell transplant (auto-HSCT) * Relapsed or refractory disease after at least 2 lines of therapy or after auto-HSCT * Relapsed or refractory disease after allogeneic hematopoietic stem cell transplant (allo-HSCT) * Relapsed or refractory B-cell ALL (patients with Burkitt's lymphoma/leukemia are not eligible) * All B-ALL patients must have detectable disease by morphology, flow cytometry, cytogenetic analysis (e.g. polymerase chain reaction \[PCR\], fluorescence in situ hybridization \[FISH\], karyotyping) or imaging (e.g. positron emission tomography \[PET\]/computed tomography \[CT\]) or a high likelihood of active disease * Refractory: failure to achieve complete response (CR) (minimal residual disease \[MRD\]-negative) at the end of induction * Relapsed: recurrence of disease after achieving CR * Evidence of CD19 expression by immunohistochemistry or flow cytometry on any prior or current tumor specimen or high likelihood of CD19 expression based on disease histology INCLUSION CRITERIA FOR LEUKAPHERESIS AND PRE-THERAPY EVALUATION * Screening evaluation appropriate for leukapheresis and T-cell collection * Adequate vascular access available or planned for leukapheresis procedure (either peripheral line or surgically placed line) * Documentation of CD19 expression on any prior or current tumor biopsy; patients who have received previous CD19-targeted therapy must have CD19-positive disease confirmed on a biopsy since completing the prior CD19-targeted therapy * Internal review of histology * Stage 2; cohort 2B (CAR-exposed) only: * Relapsed disease after achieving CR at the discretion of the principal investigator (PI) or designee, after prior CD19-targeted non-huJCAR014 CAR T-cell therapy. Patients who were in CR prior to the CD19-targeted non-huJCAR014 CAR T-cell therapy may be considered eligible at the discretion of the PI or designee. OR * Persistent disease after achieving PR at the discretion of the PI or designee, after prior CD19-targeted non-huJCAR014 CAR T-cell therapy. Patients who are less than 3 months from prior CD19-targeted non-huJCAR014 CAR T-cell therapy must have persistent disease on biopsy or imaging (e.g. PET-CT or CT) evidence of disease progression INCLUSION CRITERIA FOR LYMPHODEPLETING CHEMOTHERAPY AND huJCAR014 TREATMENT * Successful collection of T cells for huJCAR014 manufacturing * Detectable disease by imaging (for example PET +/- CT, magnetic resonance imaging \[MRI\]) and/or pathology evaluation * Karnofsky performance status \>= 60% * Assessed by the investigator to have adequate bone marrow function to receive lymphodepleting conditioning chemotherapy * Serum creatinine =\< 1.5 x age-adjusted upper limit of normal (ULN) or calculated creatinine clearance (Cockcroft and Gault) \> 30 mL/min/1.73 m\^2 * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 5 x ULN and total bilirubin \< 2.0 mg/dL unless due to malignancy or Gilbert's syndrome in the opinion of the PI or designee * Common Terminology Criteria for Adverse Events (CTCAE) grade =\< 1 dyspnea and oxygen saturation (SaO2) \>= 92% on room air * Left ventricular ejection fraction (LVEF) \>= 40% as assessed by echocardiogram (ECHO) or multiple uptake gated acquisition (MUGA) scan performed within 1 month before starting lymphodepleting chemotherapy * Women of childbearing potential (defined as all women physiologically capable of becoming pregnant) must agree to both of the following: * Use highly effective methods of contraception for at least 6 months after the last dose of huJCAR014, and * Have a negative serum pregnancy test performed within 28 days before starting lymphodepleting chemotherapy * Males who have partners of childbearing potential must agree to use an effective barrier contraceptive method for at least 6 months after the last dose of huJCAR014 Exclusion Criteria: EXCLUSION CRITERIA FOR SCREENING * For patients in stage 1 only, prior treatment with any CD19 CAR T-cell therapy is excluded * Known active hepatitis B, hepatitis C, or human immunodeficiency virus (HIV) infection * Pregnant or breastfeeding women * Any known contraindication to leukapheresis * Any known and irreversible contraindication to huJCAR014 therapy * Medical, psychological, familial, sociological, or geographical condition that does not permit compliance with the protocol as judged by the PI or designee, or unwillingness or inability to follow protocol procedures EXCLUSION CRITERIA FOR LEUKAPHERESIS AND PRE-THERAPY EVALUATION * History or presence of clinically relevant central nervous system (CNS) pathology that, in the opinion of the PI or designee, is a contraindication to lymphodepleting chemotherapy or huJCAR014 infusion * History of another primary malignancy that has not been in remission for at least 2 years with the following exceptions: nonmelanoma skin cancer, curatively treated localized prostate cancer, cervical carcinoma in situ on biopsy or a squamous intraepithelial lesion on Pap smear, or other malignancy considered by the investigator to have a low risk of relapse or progression * Active autoimmune disease requiring immunosuppressive therapy, unless considered by the PI or designee to be eligible * Presence of active acute or chronic graft versus host disease (GVHD) * Use of any of the following: * Cytotoxic or lymphotoxic agents (including prednisone \> 5 mg/day or equivalent corticosteroid) within 1 week prior to leukapheresis; physiologic corticosteroid replacement, and topical or inhaled corticosteroids are not excluded * GVHD therapies within 4 weeks prior to leukapheresis (e.g., calcineurin inhibitors, methotrexate or other chemotherapeutics, mycophenolate, rapamycin, thalidomide, immunosuppressive antibodies such as anti-TNF, anti-IL-6, or anti-IL-6R) * Experimental agents within 4 weeks prior to leukapheresis unless progression is documented on therapy and at least 3 half-lives have elapsed prior to leukapheresis * Radiation encompassing all sites of known tumor within 6 weeks prior to leukapheresis, unless there is evidence of active disease after radiation by imaging, biopsy or clinical evaluation * Allo-HSCT within 60 days prior to leukapheresis or donor lymphocyte infusion (DLI) within 6 weeks prior to leukapheresis * Treatment with cladribine within 3 months prior to leukapheresis * Treatment with alemtuzumab within 3 months prior to leukapheresis EXCLUSION CRITERIA FOR LYMPHODEPLETING CHEMOTHERAPY AND huJCAR014 TREATMENT * Uncontrolled and serious infection * Presence of active acute or chronic GVHD * DLI within 6 weeks prior to lymphodepletion chemotherapy