This phase III trial studies how well trastuzumab emtansine (T-DM1) and tucatinib work in preventing breast cancer from coming back (relapsing) in patients with high risk, HER2 positive breast cancer. T-DM1 is a monoclonal antibody, called trastuzumab, linked to a chemotherapy drug, called DM1. Trastuzumab is a form of targeted therapy because it attaches to specific molecules (receptors) on the surface of cancer cells, known as HER2 receptors, and delivers DM1 to kill them. Tucatinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Giving T-DM1 and tucatinib may work better in preventing breast cancer from relapsing in patients with HER2 positive breast cancer compared to T-DM1 alone.