Aberrant cell proliferation, a hallmark of most cancers, requires the escape from intrinsic antitumour barriers. Primary among these is the DNA damage response (DDR). In both cell culture-models and in early stages of tumorigenesis in vivo, activated oncogenes induce DNA replication stress and DNA double-strand breaks (DSBs), leading to DDR activation and p53-dependent apoptosis and/or senescence. The means by which tumour-initiating cells, also termed cancer stem cells (CSCs), circumvent this oncosuppressive response is unknown. Here we demonstrate that the ZEB1 transcription factor provides breast CSCs with the ability to withstand an aberrant mitogenic activity. Its forced expression in human mammary epithelial cells is sufficient to alleviate DNA replicative stress and to decrease the production of reactive oxygen species, an important contributor to DDR and oncogene-induced senescence. Consistently, human breast cancer cells with endogenous ZEB1 expression show two characteristic features: low levels of DSBs and DDR markers, reflecting mitigation of the DNA replication stress, and a low p53 mutation frequency, reflecting a weak selective pressure for inactivation. Using high-throughput sequencing analysis of controlled cellular models, we further demonstrate that ZEB1 delays the onset of structural chromosomal instability (CIN), a known consequence of replicative stress and prevents the emergence of chromosome 8p deletions and 8q amplifications, two prevalent abnormalities in high-grade breast cancers. Supporting these findings, ZEB1 expression discriminates human breast tumours by their copy number alterations (CNAs) and chromosome 8 aberrations. We propose that the tumorigenic potential of CSCs relies upon their unique capacity to tolerate oncogenic stimuli through the alleviation of DNA replication stress.