This phase III trial compares the effect of usual treatment with trastuzumab emtansine (T-DM1) alone vs. T-DM1 in combination with tucatinib. 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 blocks HER2, which may help keep cancer cells from growing and may kill them. Giving T-DM1 in combination with tucatinib may work better in preventing breast cancer from relapsing in patients with HER2 positive breast cancer compared to T-DM1 alone.