This phase II trial studies how well giving paclitaxel albumin-stabilized nanoparticle (Nab-paclitaxel) formulation together with bevacizumab followed by bevacizumab and erlotinib hydrochloride work in treating patients with metastatic breast cancer. Drugs used in chemotherapy, such as paclitaxel albumin-stabilized nanoparticle formulation, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. Monoclonal antibodies, such as bevacizumab, can prevent cancer growth by blocking the ability of cancer cells to grow and spread. Erlotinib hydrochloride may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. This trial evaluates a maintenance treatment with erlotinib and bevacizumab after Nab-paclitaxel and bevacizumab which may control cancer growth with biologic therapies.