This randomized phase III trial studies bevacizumab and intravenous (given into a vein) chemotherapy to see how well they work compared with bevacizumab and intraperitoneal (given into the abdominal cavity) chemotherapy in treating patients with stage II-III ovarian epithelial cancer, fallopian tube cancer, or primary peritoneal cancer. Monoclonal antibodies, such as bevacizumab, can block the ability of tumor cells to grow and spread by blocking the growth of new blood vessels necessary for tumor growth. Drugs used in chemotherapy, such as paclitaxel, carboplatin, and cisplatin, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. It is not yet known whether giving bevacizumab together with intravenous chemotherapy is more effective than giving bevacizumab together with intraperitoneal chemotherapy in treating patients with ovarian epithelial cancer, fallopian tube cancer, or primary peritoneal cancer.