The cohesin subunit STAG2 has emerged as a recurrently inactivated tumor suppressor in human cancers. Using candidate approaches, recent studies have revealed a synthetic lethal interaction between STAG2 and its paralog STAG1 To systematically probe genetic vulnerabilities in the absence of STAG2, we have performed genome-wide CRISPR screens in isogenic cell lines and identified STAG1 as the most prominent and selective dependency of STAG2-deficient cells. Using an inducible degron system, we show that chemical genetic degradation of STAG1 protein results in the loss of sister chromatid cohesion and rapid cell death in STAG2-deficient cells, while sparing STAG2-wild-type cells. Biochemical assays and X-ray crystallography identify STAG1 regions that interact with the RAD21 subunit of the cohesin complex. STAG1 mutations that abrogate this interaction selectively compromise the viability of STAG2-deficient cells. Our work highlights the degradation of STAG1 and inhibition of its interaction with RAD21 as promising therapeutic strategies. These findings lay the groundwork for the development of STAG1-directed small molecules to exploit synthetic lethality in STAG2-mutated tumors.