Genetically Engineered Cells (NY-ESO-1 TCR Engineered T Cells and HSCs) After Melphalan Conditioning Regimen in Treating Patients With Recurrent or Refractory Ovarian, Fallopian Tube, or Primary Peritoneal Cancer

Inclusion Criteria:

* Must have a diagnosis of platinum-sensitive or platinum-resistant recurrent or refractory epithelial ovarian, primary peritoneal or fallopian tube carcinoma and have progressed, relapsed, or recurred through at least one or more prior lines of standard-of-care therapies. For platinum sensitive patients, the standard of care therapies include additional platinum-containing regimens and bevacizumab
* Have been informed of other treatment options
* Must be HLA- A\*02.1 and HLA-DP\*04 positive. Retesting is not required for patients who have previous documented positivity
* Patient's tumor must be positive by histological or molecular assay for NY-ESO-1
* Have an Eastern Cooperative Oncology group (ECOG) performance status of 0 or 1
* Life expectancy of \> 4 months
* At least 4 weeks from prior chemotherapy, radiotherapy or immunotherapy, or prior investigational agents
* Must have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1
* Must have adequate venous access for apheresis (pheresis catheter placement for cell collection is allowed)
* Since the study drug may affect pregnancy since it targets proteins present during development, women of childbearing potential are requested to use acceptable methods of birth control for the duration of the study and until persistence of the study drug is no longer detected in the patient by polymerase chain reaction (PCR). This may be a period of several years. Methods for acceptable birth control include: condoms, diaphragm or cervical cap with spermicide, intrauterine device, and hormonal contraception. It is recommended that a combination of two methods be used
* Leukocytes \>= 3 x 10\^9/L
* Absolute neutrophil count \>= 1.5 x 10\^9/L
* Platelets \>= 100 x 10\^9/L
* Total bilirubin within normal institutional limits
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal
* Creatinine =\< 2 x upper limit of normal (ULN); if creatinine level \> 2 x ULN, then creatinine clearance must be \> 60 mL/min
* Patient must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure
* Participant must agree to and arrange for a caregiver (age \>= 18 years old) available 24 hours a day/ 7 days a week and arrange for lodging within 45 minutes drive to Roswell Park and transportation for a period of time after discharge from the hospital. The exact amount of time will depend on the individual status as determined by the treating physician

Exclusion Criteria:

* Patients may not be receiving any other investigational agents
* Known cases of clinically active brain metastases (brain magnetic resonance imaging \[MRI\] as clinically indicated). Prior evidence of brain metastasis successfully treated with surgery or radiation therapy will not be exclusion for participation as long as they are deemed under control at the time of study enrollment
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to agents used in the study
* Prior malignancy (except non melanoma skin cancer) within 3 years
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, or psychiatric illness/social situations that would limit compliance with study requirements
* Use of chronic corticosteroids, hydroxyurea, or immunomodulating agents (e.g., interleukin 2, interferon alpha or gamma, granulocyte colony stimulating factors, etc.) within 30 days prior to study entry

  * NOTE: Recent or current use of inhaled steroids and topical steroids are not exclusionary. If subjects are prescribed a brief course of oral corticosteroids, the use should be limited to less than 7 days. Use of steroids before apheresis and immune assessment blood draws should be discouraged as it will affect white blood cell function
* Active infection with human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), or cytomegalovirus (CMV) as defined below, due to the immunosuppressive effects of chemo-conditioning used and the unknown risks associated with viral replication

  * Positive serology for HIV
  * Active hepatitis B infection as determined by test for hepatitis B surface antigen (Ag)
  * Active hepatitis C. Patients will be screened for HCV antibody. If the HCV antibody is positive, a screening HCV ribonucleic acid (RNA) by any reverse transcriptase (RT) PCR or branched deoxyribonucleic acid (bDNA) assay must be performed at screening by a local laboratory with a Clinical Laboratory Improvement Act (CLIA) certification or its equivalent. Eligibility will be determined based on a negative screening value. The test is not required if documentation of a negative result of a HCV RNA test performed within 60 days prior to screening is provided.
  * Serology (CMV IgG) positive for active CMV
* Received any previous gene therapy using an integrating vector within 6 months
* Pregnancy or breast-feeding
* Lack of availability of a patient for immunological and clinical follow up assessment
* Evidence or history of significant cardiac disease (including myocardial infarction \[MI\] in the past 6 months, significant cardiac arrhythmia, stage III or IV congestive heart failure \[CHF\]). Cardiac stress test will be done if clinically indicated. (The specific test to be chosen at the discretion of the principal investigator \[PI\])
* Patients with pulmonary function test abnormalities as evidenced by a forced expiratory volume in 1 second (FEV1)/forced vital capacity (FVC) \< 70% of predicted for normality will be excluded