Carboplatin and Paclitaxel or Oxaliplatin and Capecitabine With or Without Bevacizumab as First-Line Therapy in Treating Patients With Newly Diagnosed Stage II-IV or Recurrent Stage I Epithelial Ovarian or Fallopian Tube Cancer

Inclusion Criteria:

* Patients with a histologic diagnosis of mucinous adenocarcinoma of the ovary or fallopian tube with either optimal (=\< 1 cm residual disease) or suboptimal residual disease following initial surgery; patients may have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 or no measurable disease
* All patients must have had appropriate surgery including appendectomy (unless patient has history of prior appendectomy) for ovarian or fallopian tube carcinoma with appropriate tissue available for histologic evaluation to confirm diagnosis and stage
* Patients must have stage II-IV disease (new or recurrent-chemonaïve; no brain metastasis) or recurrent stage I disease (chemonaïve)
* Newly diagnosed patients must begin protocol therapy within 10 weeks of primary debulking; for stage I recurrent patients (chemonaïve), they should begin protocol therapy within 14 days of randomization
* Patients must have a negative colonoscopy within 1 year of enrolling in the study
* Absolute neutrophil count (ANC) \>= 1,500/mcl
* White blood cell (WBC) count \>= 3,000/mcl
* Platelets \>= 100,000/mcl
* Hemoglobin (Hgb) \>= 10 g/dl (can be post transfusion)
* Creatinine less than or equal to 1.5 x institutional upper limit normal (ULN) OR creatinine clearance \> 50 cc/min
* Bilirubin =\< 1.5 x ULN
* Serum glutamic oxaloacetic transaminase (SGOT) =\< to 2.5 x ULN
* Alkaline phosphatase =\< to 2.5 x ULN
* Neuropathy (sensory and motor) =\< Common Terminology Criteria for Adverse Events (CTCAE) grade 1
* Urine dipstick for proteinuria \< 2+; if urine dipstick is \>= 2+, 24 hour urine must demonstrate =\< 1 g protein in 24 hours OR patients must have a urine protein-to-creatinine ratio (UPCR) \< 1.0 mg/dL
* Prothrombin time (PT) =\< 1.5 x ULN
* Activated prothrombin time (APTT) =\< 1.5 x ULN
* Patients of childbearing potential must agree to practice an effective form of birth control during study treatment and for six months after completion of treatment
* Patients who have met the pre-entry requirements
* Patients must have signed an approved informed consent and authorization permitting release of personal health information
* Patients with Gynecologic Oncology Group (GOG) performance grade of 0, 1 or 2
* Patients with life expectancy \> 3 months

Exclusion Criteria:

* Patients with known colon cancer or history of colon cancer
* Patients with primary peritoneal carcinoma
* Patients with a history of other invasive malignancies, with the exception of non-melanoma skin cancer, are excluded if there is any evidence of other malignancy being present within the last 5 years; patients are also excluded if their previous cancer treatment contraindicates this protocol therapy
* Patients who have received chemotherapy, radiotherapy or any investigational treatment for a gynecologic cancer (does include breast cancer) or colorectal cancer prior to enrollment
* Patients with a major surgical procedure anticipated during the course of the study; this includes but is not limited to: abdominal surgery (laparotomy or laparoscopy) such as colostomy or enterostomy reversal, interval or secondary cytoreductive surgery, or second look surgery; please consult with the study chair prior to patient entry for any questions related to the classification of surgical procedures
* Patients may have minor surgical procedures (i.e., mediport insertion) fine needle aspiration or core biopsies as long as it is performed \> 7 days prior to the first date of bevacizumab therapy and there is no evidence of wound disruption or impaired healing
* Patients with surgery (including open biopsy) within 4 weeks prior to anticipated first dose of bevacizumab (allowing for fact that bevacizumab can be omitted from first cycle of chemotherapy)
* Patients with a history of abdominal fistula or perforation within the past 12 months
* Patients with a current, serious, non-healing wound, ulcer, or bone fracture; patients with granulating incisions healing by secondary intention with no evidence of fascial dehiscence or infection are eligible but require weekly wound examinations
* Patients with known hypersensitivity to Chinese hamster cell products or other recombinant human or humanized antibodies
* Patients with mixed epithelial ovarian cancer histology
* Patients with tumors of low malignant potential
* History or evidence of upon physical examination of central nervous system (CNS) disease, including history of primary brain tumor or any history of brain metastases, or seizures not controlled with standard medical therapy
* Uncontrolled hypertension, defined as systolic \> 150 mm Hg or diastolic \> 100 mm Hg; patients with a history of hypertension are permitted
* Myocardial infarction or unstable angina within 12 months of the first date of bevacizumab therapy
* New York Heart Association (NYHA) grade II or greater congestive heart failure or serious cardiac arrhythmia requiring medication; women who have received prior treatment with anthracycline (including doxorubicin and/or liposomal doxorubicin) and have an ejection fraction \< 50% will be excluded from the study
* Grade 1, category 2 or greater, peripheral vascular disease; patient cannot have anything worse than mild, symptomatic claudication with exercise
* History of cerebrovascular accident (CVA, stroke), transient ischemic attack (TIA) or subarachnoid hemorrhage within six months of the first date of bevacizumab therapy
* History of pulmonary embolism or deep vein thrombosis in the past 6 months
* Previous history of malabsorption or other conditions preventing oral treatment
* Patients who are pregnant or nursing
* Patients with acute hepatitis or active infection that requires parenteral antibiotics
* Patients with active bleeding or pathologic conditions that carry a high risk of bleeding such as a known bleeding disorder, coagulopathy or tumor involving the major vessels
* Patients taking warfarin