Single-cell CRISPR screens link genetic perturbations to transcriptional states, but high-throughput methods connecting these induced changes to their regulatory foundations are limited. Here, we introduce Multiome Perturb-seq, extending single-cell CRISPR screens to simultaneously measure perturbation-induced changes in gene expression and chromatin accessibility. We apply Multiome Perturb-seq in a CRISPRi screen of 13 chromatin remodelers in human RPE-1 cells, achieving efficient assignment of sgRNA identities to single nuclei via an improved method for capturing barcode transcripts from nuclear RNA. We organize expression and accessibility measurements into coherent programs describing the integrated effects of perturbations on cell state, finding that ARID1A and SUZ12 knockdowns induce programs enriched for developmental features. Modeling of perturbation-induced heterogeneity connects accessibility changes to changes in gene expression, highlighting the value of multimodal profiling. Overall, our method provides a scalable and simply implemented system to dissect the regulatory logic underpinning cell state. A record of this paper's transparent peer review process is included in the supplemental information.