Large genomic insertions, deletions, and inversions are a potent source of functional and fitness-altering variation, but are challenging to resolve with short-read DNA sequencing alone. While recent long-read sequencing technologies have greatly expanded the catalog of structural variants (SVs), their costs have so far precluded their application at population scales. Given these limitations, the role of SVs in human adaptation remains poorly characterized. Here, we used a graph-based approach to genotype 107,866 long-read-discovered SVs in short-read sequencing data from diverse human populations. We then applied an admixture-aware method to scan these SVs for patterns of population-specific frequency differentiation—a signature of local adaptation. We identified 220 SVs exhibiting extreme frequency differentiation, including several SVs that were among the lead variants at their corresponding loci. The top two signatures traced to separate insertion and deletion polymorphisms at the immunoglobulin heavy chain locus, together tagging a 325 Kbp haplotype that swept to high frequency and was subsequently fragmented by recombination. Alleles defining this haplotype are nearly fixed (60-95%) in certain Southeast Asian populations, but are rare or absent from other global populations composing the 1000 Genomes Project. Further investigation revealed that the haplotype closely matches with sequences observed in two of three high-coverage Neanderthal genomes, providing strong evidence of a Neanderthal-introgressed origin. This extraordinary episode of positive selection, which we infer to have occurred between 1700 and 8400 years ago, corroborates the role of immune-related genes as prominent targets of adaptive archaic introgression. Our study demonstrates how combining recent advances in genome sequencing, genotyping algorithms, and population genetic methods can reveal signatures of key evolutionary events that remained hidden within poorly resolved regions of the genome.