A Phase III Randomized Controlled Clinical Trial of Carboplatin and Paclitaxel (or Gemcitabine) Alone or in Combination With Bevacizumab (NSC #704865) Followed by Bevacizumab and Secondary Cytoreductive Surgery in Platinum-Sensitive, Recurrent Ovarian, Peritoneal Primary and Fallopian Tube Cancer. NCI-Supplied Agents: Bevacizumab (NSC #704865)

• Patients enrolled after August 28, 2011 must be candidates for cytoreductive surgery and consent to have their surgical treatment determined by randomization

• Patients must have histologic diagnosis of epithelial ovarian carcinoma, peritoneal primary or fallopian tube carcinoma, which is now recurrent

• Patients with the following histologic epithelial cell types are eligible: serous adenocarcinoma, endometrioid adenocarcinoma, mucinous adenocarcinoma, undifferentiated carcinoma, clear cell adenocarcinoma, mixed epithelial carcinoma, transitional cell carcinoma, malignant Brenner's tumor, or adenocarcinoma not otherwise specified (N.O.S.)

• Patients must have had a complete response to front-line platinum-taxane therapy (at least three cycles)

• A complete response to front-line chemotherapy must include: negative physical exam, negative pelvic exam and normalization of CA125, if elevated at baseline; although not required, any radiographic assessment of disease status (e.g. CT, magnetic resonance imaging \[MRI\], positron emission tomography \[PET\]/CT, etc) obtained following the completion of primary therapy should be considered negative for disease

• All patients must have also had a treatment-free interval without clinical evidence of progressive disease of at least 6 months from completion of front-line chemotherapy (both platinum and taxane); front-line therapy may have included a biologic agent (i.e. bevacizumab)

• Front-line treatment may include maintenance therapy following complete clinical or pathological response; however, maintenance cytotoxic chemotherapy must be discontinued for a minimum of 6 months prior to documentation of recurrent disease; patients receiving maintenance biological therapy or hormonal therapy are ELIGIBLE provided their recurrence is documented more than 6 months from primary cytotoxic chemotherapy completion (includes maintenance chemotherapy) AND a minimum 4 weeks has elapsed since their last infusion of biological therapy

• Patients must have clinically evident recurrent disease for the purpose of this study

• Measurable disease (Response Evaluation Criteria in Solid Tumors \[RECIST\]) is defined as at least one lesion that can be accurately measured in at least one dimension (longest dimension to be recorded); each lesion must be more than or equal to 20 mm when measured by conventional techniques, MRI or CT, or more than or equal to 10 mm when measured by spiral CT

• Absolute neutrophil count (ANC) greater than or equal to 1,500/mm\^3, equivalent to Common Toxicity Criteria for Adverse Events version (v)4.0 (CTCAE) grade 1

• Platelets greater than or equal to 100,000/mm\^3 (CTCAE grade 0-1)

• Creatinine (non-isotope dilution mass spectrometry \[IDMS\]) =\< 1.5 x institutional upper limit normal (ULN), CTCAE grade 1

• Total bilirubin =\< 1.5 ULN (CTCAE grade 1)

• Serum glutamic oxaloacetic transaminase (SGOT)/aspartate aminotransferase (AST) =\< 2.5 times the upper limit of normal in the absence of liver metastasis; SGOT/AST \< 5.0 times ULN in the presence of liver metastasis

• Alkaline phosphatase =\< 2.5 times the upper limit of normal in the absence of liver metastasis; alkaline phosphatase \< 5.0 times ULN in the presence of liver metastasis

• This criterion applies only to the patients enrolled before August 29, 2011 and those enrolled after this date electing to receive bevacizumab; patients must have a urine protein-to-creatinine ratio (UPCR) \< 1.0 mg/dL

• This eligibility criterion does not apply to patients enrolled after August 28, 2011; patients who are not candidates for surgical cytoreduction are eligible for the chemotherapy randomization; patients are not considered candidates for surgical cytoreduction if complete cytoreduction in the estimation of the investigator is impossible or a medical infirmity precludes exploration and debulking

• Patients must have met the pre-entry requirements as specified

• Patients must have signed an approved informed consent and authorization permitting release of personal health information

• Patients must have a Gynecologic Oncology Group (GOG) performance status of 0, 1, or 2