Large genomic insertions and deletions are a potent source of functional variation, but are challenging to resolve with short-read sequencing, limiting knowledge of the role of such structural variants (SVs) in human evolution. Here, we used a graph-based method to genotype long-read-discovered SVs in short-read data from diverse human genomes. We then applied an admixture-aware method to identify 220 SVs exhibiting extreme patterns of frequency differentiation – a signature of local adaptation. The top two variants traced to the immunoglobulin heavy chain locus, tagging a haplotype that swept to near fixation in certain southeast Asian populations, but is rare in other global populations. Further investigation revealed evidence that the haplotype traces to gene flow from Neanderthals, corroborating the role of immune-related genes as prominent targets of adaptive introgression. Our study demonstrates how recent technical advances can help resolve signatures of key evolutionary events that remained obscured within technically challenging regions of the genome.