This phase Ib/II trial studies side effects and best dose of inotuzumab ozogamicin and how well it works when given together with vincristine sulfate liposome in treating patients with CD22 positive (+) B-cell acute lymphoblastic leukemia that has come back or dose not respond to treatment. Inotuzumab ozogamicin is a monoclonal antibody, called inotuzumab, linked to a toxic agent called ozogamicin. Inotuzumab attaches to CD22+ cancer cells in a targeted way and delivers ozogamicin to kill them. Drugs used in chemotherapy, such as vincristine sulfate liposome, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving inotuzumab ozogamicin and vincristine sulfate liposome together may work better in treating patients with CD22+ B-cell acute lymphoblastic leukemia compared to giving inotuzumab ozogamicin or vincristine sulfate liposome alone.