This phase II trial studies how well atezolizumab and cobimetinib work in treating patients with non-small cell lung cancer that has spread from where it first started (primary site) to other places in the body (metastatic), has come back (recurrent), or does not respond to treatment (refractory). Immunotherapy with monoclonal antibodies, such as atezolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Cobimetinib is used in patients whose cancer has a mutated (changed) form of a gene called BRAF. It is in a class of medications called kinase inhibitors. It works by blocking the action of an abnormal protein that signals cancer cells to multiply. This helps slow or stop the spread of cancer cells. Giving atezolizumab and cobimetinib may work better in treating patients with non-small cell lung cancer.