Paclitaxel and Carboplatin With or Without Bevacizumab in Treating Patients With Stage II, Stage III, or Stage IV Ovarian Epithelial Cancer, Primary Peritoneal Cancer, or Fallopian Tube Cancer

Inclusion Criteria: * Primary Surgery and Neoadjuvant Chemotherapy with Interval Cytoreductive Surgery Patients: * Patients must have measurable disease; at least one target lesion must have a minimum length of 1 cm in both the long and short axis (determined at the local site); for primary surgery patients, if no radiographic evidence of measurable disease is obtained prior to registration this can be based on surgical findings; imaging then would need to be completed in the 14 days between Gynecology Oncology Group (GOG) registration and chemotherapy initiation * After GOG registration, the American College of Radiology \[ACR\] Imaging Core Laboratory will confirm target lesion as required per protocol; the GOG-eligibility (RECIST) scan and baseline T0 perfusion CT scans will be reviewed prior to the intermediate T1 perfusion CT time point * Primary Surgery Patients: * Patients with a histologic diagnosis of epithelial ovarian cancer, peritoneal primary carcinoma or fallopian tube cancer, stage II -IV suboptimally debulked (any residual disease \> 1 cm); International Federation of Gynecology and Obstetrics (FIGO) stage is assessed following the completion of initial abdominal surgery, appropriate imaging studies and with appropriate tissue available for histologic evaluation; the minimum surgery required is an abdominal surgery providing tissue for histologic evaluation and establishing and documenting the primary site and stage; if additional surgery was performed, it should have been in accordance with appropriate surgery for ovarian or peritoneal carcinoma described in the GOG Surgical Procedures Manual * Neoadjuvant Chemotherapy (NAC) with Interval Cytoreductive Surgery (ICS) Patients: * For patients undergoing NAC-ICS, a core tissue (not fine needle aspiration) biopsy is required; the tissue must be consistent with a Müllerian origin; patients will require documentation of at least stage II or extraovarian sites of disease acquired via imaging or surgery (without attempt at cytoreduction) * Patients with the following histologic epithelial cell types are eligible: serous, endometrioid, clear cell, mucinous adenocarcinoma, undifferentiated carcinoma, mixed epithelial carcinoma, transitional cell carcinoma, malignant Brenner's Tumor, or adenocarcinoma not otherwise specified (N.O.S.); however, the histologic features of the tumor must be compatible with a primary Müllerian epithelial adenocarcinoma; patients may have co-existing fallopian tube carcinoma in-situ so long as the primary origin of invasive tumor is ovarian, peritoneal or fallopian tube; of note, patients with clear cell and mucinous tumors will be eligible unless there is a higher priority protocol * Absolute neutrophil count (ANC) greater than or equal to 1,500/mcl; this ANC cannot have been induced or supported by granulocyte colony stimulating factors * Platelets greater than or equal to 100,000/mcl * Creatinine =\< 1.5 x institutional upper limit normal (ULN) * Bilirubin less than or equal to 1.5 x ULN * Serum glutamic oxaloacetic transaminase (SGOT) less than or equal to 3 x ULN * Alkaline phosphatase less than or equal to 2.5 x ULN * Neuropathy (sensory or motor) less than or equal to Common Terminology Criteria for Adverse Events (CTCAE) grade 1 * Patients must have a GOG performance status of 0, 1, or 2 * Patients must be entered within 12 weeks of diagnostic/staging surgery * Patients who have met the pre-entry requirements * An approved informed consent and authorization permitting release of personal health information and must be signed by the patient or guardian * Only applies for patients who elect to receive bevacizumab: * Patients in this trial may receive ovarian estrogen +/- progestin replacement therapy as indicated at the lowest effective dose(s) for control of menopausal symptoms at any time, but not high-dose progestins for management of anorexia while on protocol-directed therapy or prior to disease progression due to thrombophlebitis risk * Prothrombin time (PT) such that international normalized ratio (INR) is =\< 1.5 (or an in-range INR, usually between 2 and 3, if a patient is on a stable dose of therapeutic warfarin for management of venous thrombosis including pulmonary thromboembolus) and a partial prothrombin time (PTT) \< 1.2 times the upper limit of normal; (heparin, lovenox or alternative anticoagulants are acceptable) * All patients enrolled into GOG-0262 at sites where ACRIN 6695 is open will be enrolled in the advanced imaging protocol; patients receiving adjuvant or neoadjuvant chemotherapy are eligible for ACRIN 6695; the following sentence does not apply to those patients entered after 02/08/2012: if a patient declines to participate in the perfusion imaging portion of the protocol, a clinical rationale for declination of imaging form will be completed as part of the data submission for ACRIN 6695 * ACRIN 6695 Eligible Patients: * Confirmation of ACRIN 6695 eligibility after the baseline T0 perfusion computed tomography (CT) will be assessed by the ACR Imaging Core Lab: At least one target lesion must have a minimum length of 1 cm in both the long and short axis (as determined by the local site), at least half of the target lesion must have attenuation greater than or equal to 10 Hounsfield Units (HU) on the unenhanced CT, and at least half of the lesion must have maximum enhancement greater than or equal to 5 HU in the perfusion CT scan (as determined by the ACR Imaging Core Lab) Exclusion Criteria: * Patients with a current diagnosis of borderline epithelial ovarian tumor (formerly "tumors of low malignant potential") or recurrent invasive epithelial ovarian, primary peritoneal or fallopian tube cancer treated with surgery only (such as patients with stage I-A or I-B low grade epithelial ovarian or fallopian tube cancers) are not eligible; patients with a prior diagnosis of a borderline tumor that was surgically resected and who subsequently develop an unrelated, new invasive epithelial ovarian, peritoneal primary or fallopian tube cancer are eligible, provided that they have not received prior chemotherapy for any ovarian tumor * Patients who have received prior radiotherapy to any portion of the abdominal cavity or pelvis; prior radiation for localized cancer of the breast, head and neck, or skin is permitted, provided that it was completed more than three years prior to registration, and the patient remains free of recurrent or metastatic disease * Patients who have received prior chemotherapy for any abdominal or pelvic tumor including neo-adjuvant chemotherapy for their ovarian, primary peritoneal or fallopian tube cancer; patients may have received prior adjuvant chemotherapy for localized breast cancer, provided that it was completed more than three years prior to registration, and that the patient remains free of recurrent or metastatic disease * Patients who have received any targeted therapy (including but not limited to vaccines, antibodies, tyrosine kinase inhibitors) or hormonal therapy for management of their epithelial ovarian, fallopian tube or peritoneal primary cancer * Patients with synchronous primary endometrial cancer, or a past history of primary endometrial cancer, unless all of the following conditions are met: stage not greater than I-A, grade 1 or 2, no more than superficial myometrial invasion, without vascular or lymphatic invasion; no poorly differentiated subtypes, including papillary serous, clear cell or other FIGO grade 3 lesions * With the exception of non-melanoma skin cancer, patients with other invasive malignancies who had (or have) any evidence of the other cancer present within the last five years or whose previous cancer treatment contraindicates this protocol therapy * Patients with acute hepatitis or active infection that requires parenteral antibiotics * Patients with clinically significant cardiovascular disease; this includes: * Myocardial infarction or unstable angina \< 6 months prior to registration * New York Heart Association (NYHA) grade II or greater congestive heart failure * Serious cardiac arrhythmia requiring medication; this does not include asymptomatic, atrial fibrillation with controlled ventricular rate * Patients who are pregnant or nursing; patients of childbearing potential must agree to use contraceptive measures during study therapy and for at least six months after completion of bevacizumab therapy * Patients who have received prior therapy with any anti-vascular endothelial growth factor (VEGF) drug, including bevacizumab * Patients with medical history or conditions not otherwise previously specified which in the opinion of the investigator should exclude participation in this study; the investigator should feel free to consult the Statistical and Data Center (SDC) randomization desk for uncertainty in this regard * Patients with known allergy to cremophor or polysorbate 80 * Only applies to patients who elect to receive bevacizumab: * Patients with serious non-healing wound, ulcer, or bone fracture; this includes history of abdominal fistula, gastrointestinal perforation or intra-abdominal abscess within 28 days; 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 active bleeding or pathologic conditions that carry high risk of bleeding, such as known bleeding disorder, coagulopathy, or tumor involving major vessels * Patients with history or evidence upon physical examination of central nervous system (CNS) disease, including primary brain tumor, seizures not controlled with standard medical therapy, any brain metastases, or history of cerebrovascular accident (CVA, stroke), transient ischemic attack (TIA) or subarachnoid hemorrhage within six months of the first date of treatment on this study * Patients with CTCAE grade 2 or greater peripheral vascular disease (at least brief \[\< 24 hours\] episodes of ischemia managed non-surgically and without permanent deficit) * Patients with a history of CVA within six months * Patients with known hypersensitivity to Chinese hamster ovary cell products or other recombinant human or humanized antibodies * Patients with clinically significant proteinuria; urine protein should be screened by urine protein-creatinine ratio (UPCR); the UPCR has been found to correlate directly with the amount of protein excreted in a 24 hour urine collection; specifically, a UPCR of 1.0 is equivalent to 1.0 gram of protein in a 24-hour urine collection; obtain at least 4 ml of a random urine sample in a sterile container (does not have to be a 24-hour urine); send sample to lab with request for urine protein and creatinine levels (separate requests); the lab will measure protein concentration (mg/dL) and creatinine concentration (mg/dL); the UPCR is derived as follows: protein concentration (mg/dL)/creatinine (mg/dL); patients must have a UPCR \< 1.0 to allow participation in the study * Patients with or with anticipation of invasive procedures as defined below: * Major surgical procedure, open biopsy or significant traumatic injury within 28 days prior to the first date of bevacizumab therapy (cycle 2) * Major surgical procedure anticipated during the course of the study; this includes, but is not limited to abdominal surgery (laparotomy or laparoscopy) prior to disease progression such as colostomy or enterostomy reversal, secondary cytoreductive surgery, or second look surgery; please consult with the SDC Randomization Desk prior to patient entry for any questions related to the classification of surgical procedures * Any tissue biopsy, such as a core biopsy, within 7 days prior to the first date of bevacizumab therapy (cycle 2) * Patients with clinical symptoms or signs of gastrointestinal obstruction and who require parenteral hydration and/or nutrition * Patients with metastasis tumor in the parenchyma of the liver or lungs with proximity to large vessels which could make the patients at high risk of lethal hemorrhage during treatment with bevacizumab (ie. hemoptysis, liver rupture) * ACRIN 6695 Ineligible Patients: * Patients with contraindication to iodinated contrast for perfusion CT imaging * Patients who receive Metformin within 48 hours before perfusion CT imaging