Genome-wide epigenomic maps have revealed millions of putative enhancers and promoters, but experimental validation of their function and high-resolution dissection of their driver nucleotides remain limited. Here, we present HiDRA (High-resolution Dissection of Regulatory Activity), a combined experimental and computational method for high-resolution genome-wide testing and dissection of putative regulatory regions. We test ~7 million accessible DNA fragments in a single experiment, by coupling accessible chromatin extraction with self-transcribing episomal reporters (ATAC-STARR-seq). By design, fragments are highly overlapping in densely-sampled accessible regions, enabling us to pinpoint driver regulatory nucleotides by exploiting differences in activity between partially-overlapping fragments using a machine learning model (SHARPR-RE). In GM12878 lymphoblastoid cells, we find ~65,000 regions showing enhancer function, and pinpoint ~13,000 high-resolution driver elements. These are enriched for regulatory motifs, evolutionarily-conserved nucleotides, and disease-associated genetic variants from genome-wide association studies. Overall, HiDRA provides a high-throughput, high-resolution approach for dissecting regulatory regions and driver nucleotides.