In cancer, linking epigenetic alterations to drivers of transformation has been difficult, in part because DNA methylation analyses must capture epigenetic variability, which is central to tumour heterogeneity and tumour plasticity. Here, by conducting a comprehensive analysis, based on information theory, of differences in methylation stochasticity in samples from patients with paediatric acute lymphoblastic leukaemia (ALL), we show that ALL epigenomes are stochastic and marked by increased methylation entropy at specific regulatory regions and genes. By integrating DNA methylation and single-cell gene-expression data, we arrived at a relationship between methylation entropy and gene-expression variability, and found that epigenetic changes in ALL converge on a shared set of genes that overlap with genetic drivers involved in chromosomal translocations across the disease spectrum. Our findings suggest that an epigenetically driven gene-regulation network, with UHRF1 (ubiquitin-like with PHD and RING finger domains 1) as a central node, links genetic drivers and epigenetic mediators in ALL.