Non-native species are threatening ecosystems and biodiversity worldwide. High genetic variation is thought to be a critical factor for invasion success. The global invasion of a few clonal lineages of the gastropod Potamopyrgus antipodarum is thus both puzzling and has the potential to help illuminate why some invasions succeed while others fail. Here, we used SNP markers and a geographically broad sampling scheme (N = 1617) including native New Zealand populations and invasive North American and European populations to provide the first widescale population genetic assessment of the relationships between and amongst native and invasive P. antipodarum. We used a combination of traditional and Bayesian molecular analyses to demonstrate that New Zealand populations harbor very high diversity relative to the invasive populations and are the source of the two main European genetic lineages. One of these two European lineages was in turn the source of at least one of the two main North American genetic clusters of invasive P. antipodarum, located in Lake Ontario. The other widespread North American group had a more complex origin that included the other European lineage and two New Zealand clusters. Altogether, our analyses suggest that just a small handful of clonal lineages of P. antipodarum were responsible for invasion across continents. Our findings provide critical information for prevention of additional invasions and control of existing invasive populations and are of broader relevance towards understanding the establishment and evolution of asexual populations and the forces driving biological invasion.