This phase II trial studies how well trastuzumab emtansine works in treating older patients with human epidermal growth factor receptor 2 (HER2)-positive stage I-III breast cancer. HER2 is a protein found on the surface of cancer cells that helps them to grow and spread. Trastuzumab emtansine may kill cancer cells by binding to HER2-positive on the surface of the tumor cells and blocking their ability grow and spread.