Fludarabine and Cyclophosphamide With or Without Rituximab Before CD19 Chimeric Antigen Receptor T Cells for the Treatment of Relapsed or Refractory Diffuse Large B-Cell Lymphoma

Inclusion Criteria: * Provision of signed and dated informed consent form * Stated willingness to comply with all study procedures and availability for the duration of the study * Commercial CD19 CAR T cell product not available for the patient * Male or female, aged \>= 18 * In good general health as evidenced by medical history or as determined by the principal investigator (PI) * Ability to swallow oral medication and willingness to adhere to the study intervention and any required medications * For females of reproductive potential: use of highly effective contraception (oral contraceptives, intrauterine device) during screening confirmed with serum pregnancy test, and agreement to use such a method during study participation and for an additional 4 weeks after the end of CD19 CAR T cell infusion * For males of reproductive potential: use of condoms or other methods to ensure effective contraception with partner * Agreement to adhere to lifestyle considerations throughout study duration including abstaining from tobacco and drug use * Subjects must have relapsed or refractory diffuse large B cell lymphoma treated with at least two lines of therapy Subjects must have failed to have a complete response, or have recurrent disease after the last treatment regimen. Subjects must have previously been treated with a regimen that includes an anthracycline and an anti-CD20 monoclonal antibody. Autologous transplant will be counted as one line of therapy * (CNS cohort) SSubjects must have primary or secondary CNS lymphoma and must fail to achieve a complete response (refractory disease), have progressive disease, or relapsed disease per the International Primary CNS Lymphoma Collaborative Group (IPCG) criteria following at least one prior line of therapy. First-line therapies include high dose methotrexate-based therapy but may also include temozolomide, high dose cytarabine,, lenalidomide, ibrutinib and rituximab. Radiation therapy, lenalidomide monotherapy and ibrutinib monotherapy are considered first line therapy if patient was not eligible for methotrexate-based chemotherapy at time of initial treatment but now meets study eligibility criteria * The patient's disease must be CD19 positive, either by immunohistochemistry or flow cytometry analysis on the last biopsy available * Age \>= 18 years * Performance status: Adult Subjects: Eastern Cooperative Oncology Group (ECOG) \>= 1; Subjects \> 10 years of age: Karnofsky \>= 80%; For CNS cohort, ECOG ≥ 2. * Absolute neutrophil count (ANC) \>= 1000 * Platelets \>= 100/mm\^3 * Hemoglobin \> 8 g/dL * ANC \>= 500 is acceptable if documented bone marrow involvement by disease * Creatinine clearance (estimated by Cockcroft Gault) or using 24 hour (hr) urine collection \>= 50 cc/min * Total bilirubin =\< 2 mg/dL except in subjects with Gilbert's Syndrome in whom total bilirubin must be =\< 3.0 * Alanine transaminase (alanine aminotransferase \[ALT\]/serum glutamic pyruvic transaminase \[SGPT\]) and aspartate aminotransferase (aspartate aminotransferase \[AST\]/serum glutamic oxaloacetic transaminase \[SGOT\]) =\< 3 x the upper limit of normal or =\< 5 x the upper limit of normal if documented liver involvement by disease * Cardiac left ventricular ejection fraction \>= 45% as determined by an echocardiogram and no clinically significant electrocardiogram (ECG) findings * Baseline oxygen saturation \> 92% on room air * Prior cancer directed therapy wash-out: at least 2 weeks or 5 half-lives, whichever is shorter must have elapsed since any prior systemic therapy at the time the subject is planned for leukapheresis, except for radiotherapy within 10 days of apheresis, systemic corticosteroid use within 7 days of apheresis (with the exception of single dose for an allergic reaction), or any other immunosuppressive therapies within 7 days * No use of lymphodepleting agents including alemtuzumab and antithymocyte globulin for 7 days prior to peripheral blood collection, 5 days prior to CD19 CAR T cell infusion and for 90 days after infusion Exclusion Criteria: * Presence of supplemental oxygen, cardiac pacemaker * Known allergic reactions to components of the anti-CD19 CAR T cell product as evidenced by prior documented anaphylactic reaction or other clinical signs and/or symptoms of an allergic reaction as determined by the PI * Febrile illness within 3 days of admission for lymphodepleting conditioning therapy * Treatment with another investigational drug or other investigational intervention within 2 weeks of apheresis * Primary immunodeficiency * History of autoimmune diseases (ex: Crohn's, rheumatoid arthritis, systemic lupus erythematosus, Sjogren's) resulting in end organ damage or requiring systemic immunosuppressive or systemic disease modifying agents within the last two years prior to enrollment * Autologous transplant within 6 weeks and allogeneic transplant within 3 months of planned CAR T cell infusion * Recipient of CD19 CAR T cell therapy outside of this protocol * Active central nervous system or meningeal involvement by tumor. Subjects with untreated brain metastases/central nervous system (CNS) disease will be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. Patients with a history of CNS or meningeal involvement must be in a documented remission by cerebrospinal fluid (CSF) evaluation and contrast-enhanced magnetic resonance imaging (MRI) for at least 30 days prior to study enrollment * History of active malignancy other than non-melanoma skin cancer, carcinoma in situ (e.g. cervix, bladder, breast) * Active human immunodeficiency virus (HIV) infection documented by positive viral load. HIV-positive patients with undetectable viral load are not excluded. * Subjects with uncontrolled concurrent illness including, but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, psychiatric illness, or social situations that would limit compliance with study requirements * Pregnant or breastfeeding women are excluded from this study because CAR T cell therapy may be associated with the potential for teratogenic or abortifacient effects. Subjects of child-bearing or child-fathering potential must be willing to practice birth control from the time of enrollment on this study and for four (4) weeks after receiving the CAR-T cell infusion * Diagnosis of myelodysplasia on any bone marrow biopsy prior to initiation of therapy * Serologic status reflecting active hepatitis B or C infection. Patients that are positive for hepatitis B core antibody, hepatitis B surface antigen (HBsAg), or hepatitis C antibody must have a negative polymerase chain reaction (PCR) prior to enrollment. (PCR positive patients will be excluded)