This phase III trial studies how well inotuzumab ozogamicin and post-induction chemotherapy work in treating patients with high-risk B-cell lymphoblastic lymphoma (B-ALL), mixed phenotype acute leukemia, and B-lymphoblastic lymphoma (B-LLy). Inotuzumab ozogamicin is a monoclonal antibody, called inotuzumab, linked to a toxic agent called calicheamicin. Inotuzumab attaches to cancer cells in a targeted way and delivers calicheamicin to kill them. Drugs used in chemotherapy, such as cyclophosphamide, cytarabine, doxorubicin, daunorubicin, methotrexate, leucovorin, mercaptopurine, thioguanine, vincristine, and pegaspargase, 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.