The cohesin complex plays an essential role in chromosome maintenance and transcriptional regulation. Recurrent somatic mutations in the cohesin complex are frequent genetic drivers in cancer including myelodysplatic syndromes (MDS) and acute myeloid leukemia (AML). Here, using genetic dependency screens of STAG2-mutant AML, we identified DNA damage repair and replication as genetic dependencies in cohesin-mutant cells. We demonstrated increased levels of DNA damage and sensitivity of cohesin-mutant cells to PARP inhibition. We developed a mouse model of MDS in which Stag2 mutations arise as clonal secondary lesions in the background of clonal hematopoiesis driven by Tet2 mutations, and demonstrated selective depletion of cohesin-mutant cells with PARP inhibition in vivo. Finally, we demonstrated a shift from STAG2- to STAG1-containing cohesin complexes in cohesin-mutant cells, which is associated with longer DNA loop extrusion, more intermixing of chromatin compartments, and increased interaction with PARP and RPA proteins. Our findings inform the biology and therapeutic opportunities for cohesin-mutant malignancies.