Rewiring of cellular programmes in malignant cells generates cancer-specific vulnerabilities. Here, using an unbiased screening strategy aimed at identifying non-essential genes required by tumour cells to sustain unlimited proliferative capacity, we identify the male-specific lethal (MSL) acetyltransferase complex as a vulnerability of genetically unstable cancers. We find that disruption of the MSL complex and consequent loss of the associated H4K16ac mark do not substantially alter transcriptional programmes but compromise chromosome integrity and promote chromosomal instability (CIN) that progressively exhausts the proliferative potential of cancer cells through a p53-independent mechanism. This effect is dependent on pre-existing genomic instability, and normal cells are insensitive to MSL disruption. Using cell- and patient-derived xenografts from multiple cancer types, we show that excessive CIN induced by MSL disruption inhibits tumour maintenance. Our findings suggest that targeting MSL may be a valuable means to increase CIN beyond the level tolerated by cancer cells without inducing severe adverse effects in normal tissues.