Invasive species offer outstanding opportunities to identify the genomic sources of variation that contribute to rapid adaptation, as well as the genetic mechanisms facilitating invasions. The Eurasian plant yellow starthistle (Centaurea solstitialis) is highly invasive in North and South American grasslands and known to have evolved increased growth and reproduction during invasion. Here we develop new genomic resources for C. solstitialis and map the genetic basis of invasiveness traits. We present a chromosome-scale (1N = 8) reference genome using PacBio CLR and Dovetail Omni-C technologies, and functional gene annotation using RNAseq. We find repeat structure typical of the family Asteraceae, with over 25% of gene content derived from ancestral whole genome duplications (paleologs). Using an F2 mapping population derived from a cross between native and invading parents, with a restriction site-associated DNA (RAD)-based genetic map, we validate the assembly and identify 13 QTL underpinning size traits that have evolved during invasion. We find evidence that large effect QTL may be associated with structural variants between native and invading genotypes, including a variant with an overdominant and pleiotropic effect on key invader traits. We also find evidence of significant paleolog enrichment under two QTL. Our results add to growing evidence of the importance of structural variants in evolution, and to understanding of the rapid evolution of invaders.