This randomized phase II trial studies how well erlotinib hydrochloride (Tarceva) with or without bevacizumab (Avastin) works in treating patients with stage IV non-small cell lung cancer (NSCLC) with epidermal growth factor receptor (EGFR) mutations. Erlotinib hydrochloride may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Monoclonal antibodies, such as bevacizumab, may block tumor growth in different ways by targeting certain cells. Bevacizumab may also stop the growth of NSCLC by blocking the growth of new blood vessels necessary for tumor growth. It is not yet known whether erlotinib hydrochloride is more effective when given alone or with bevacizumab in treating patients with NSCLC.