Single-nucleus assay for transposase-accessible chromatin using sequencing (snATAC-seq) creates new opportunities to dissect cell type-specific mechanisms of complex diseases. Since pancreatic islets are central to type 2 diabetes (T2D), we profiled 15,298 islet cells by using combinatorial barcoding snATAC-seq and identified 12 clusters, including multiple alpha, beta and delta cell states. We cataloged 228,873 accessible chromatin sites and identified transcription factors underlying lineage- and state-specific regulation. We observed state-specific enrichment of fasting glucose and T2D genome-wide association studies for beta cells and enrichment for other endocrine cell types. At T2D signals localized to islet-accessible chromatin, we prioritized variants with predicted regulatory function and co-accessibility with target genes. A causal T2D variant rs231361 at the KCNQ1 locus had predicted effects on a beta cell enhancer co-accessible with INS and genome editing in embryonic stem cell-derived beta cells affected INS levels. Together our findings demonstrate the power of single-cell epigenomics for interpreting complex disease genetics.