The molecular control of epigenetic information relies on hundreds of proteins of diverse function, which cooperate in defining chromatin structure and DNA methylation landscapes. While many individual pathways have been characterized, how different classes of epigenetic regulators interact to build a resilient epigenetic regulatory network (ERN) remains poorly understood. Here, we show that most individual regulators are dispensable for somatic cell fitness, and that robustness emerges from multiple layers of functional cooperation and degeneracy among network components. By disrupting 200 epigenetic regulator genes, individually or in combination, we generated network-wide maps of functional interactions for representative regulators. We found that paralogues represent only a first layer of functional compensation within the ERN, with intra- or inter-class interactions buffering the effects of perturbation in a gene-specific manner: while CREBBP cooperates with multiple acetyltransferases to form a subnetwork that ensures robust chromatin acetylation, ARID1A interacts with regulators from across all functional classes. When combined with oncogene activation, the accumulated epigenetic disorder exposes a synthetic fragility and broadly sensitizes ARID1A-deficient cells to further perturbation. Our findings reveal homeostatic mechanisms through which the ERN sustains somatic cell fitness and uncover how the network remodels as the epigenome is progressively deregulated in disease.