This phase I trial studies the side effects and best dose of veliparib when given together with carboplatin, paclitaxel, and bevacizumab in treating patients with newly diagnosed stage II-IV ovarian epithelial, fallopian tube, or primary peritoneal cancer. Veliparib may stop the growth of tumor cells by blocking some of the enzymes needed for cells to repair themselves from damage and survive. Drugs used in chemotherapy, such as carboplatin and paclitaxel, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Bevacizumab, a type of drug called a monoclonal antibody, blocks tumor growth by targeting certain cells and preventing the growth of new blood vessels that tumors need to grow. Giving veliparib together with carboplatin, paclitaxel, and bevacizumab may kill more tumor cells.