CRISPR screens have been used to connect genetic perturbations with changes in gene expression and phenotypes. Here we describe a CRISPR-based, single-cell combinatorial indexing assay for transposase-accessible chromatin (CRISPR–sciATAC) to link genetic perturbations to genome-wide chromatin accessibility in a large number of cells. In human myelogenous leukemia cells, we apply CRISPR–sciATAC to target 105 chromatin-related genes, generating chromatin accessibility data for ~30,000 single cells. We correlate the loss of specific chromatin remodelers with changes in accessibility globally and at the binding sites of individual transcription factors (TFs). For example, we show that loss of the H3K27 methyltransferase EZH2 increases accessibility at heterochromatic regions involved in embryonic development and triggers expression of genes in the HOXA and HOXD clusters. At a subset of regulatory sites, we also analyze changes in nucleosome spacing following the loss of chromatin remodelers. CRISPR–sciATAC is a high-throughput, single-cell method for studying the effect of genetic perturbations on chromatin in normal and disease states.