Ductal carcinoma in situ (DCIS) is an early-stage breast cancer that infrequently progresses to invasive ductal carcinoma (IDC). Genomic evolution has been difficult to delineate during invasion due to intratumor heterogeneity and the low number of tumor cells in the ducts. To overcome these challenges, we developed Topographic Single Cell Sequencing (TSCS) to measure genomic copy number profiles of single tumor cells while preserving their spatial context in tissue sections. We applied TSCS to 1,293 single cells from 10 synchronous patients with both DCIS and IDC regions in addition to exome sequencing. Our data reveal a direct genomic lineage between in situ and invasive tumor subpopulations and further show that most mutations and copy number aberrations evolved within the ducts prior to invasion. These results support a multiclonal invasion model, in which one or more clones escape the ducts and migrate into the adjacent tissues to establish the invasive carcinomas.