Atlantic lionfish have caused measurable ecosystem damage in their invaded range. As a matter of urgency, strategies to suppress lionfish have therefore risen to the fore amongst the ocean management community. We use a biophysical model and ocean climate data to demonstrate how Atlantic lionfish larvae are dispersed by currents and how this dispersal, combined with their breeding strategy, negates effective control using methods traditionally executed on a local scale. This study quantitatively emphasizes the high level of larval connectivity that exists between the many nations whose waters now support established lionfish populations. For any given area, our results indicate that the key to suppressing the invasion is to simultaneously choke all upstream linkages that supply external larvae and renourish the local population. On the basis of a case study developed for the Carolinas, USA, an area of high lionfish abundance, the model suggests that such a strategy requires monthly culls that remove 20% of lionfish biomass in the Carolinas and all locations to which the Carolinas are linked hydrographically. Conversely, if culls target only those locales that deliver the majority (95%) of lionfish larvae to the Carolinas, and therefore ignore those locations which contribute the remaining 5% of lionfish larvae, the requisite cull rate exceeds 60% and suppresses lionfish abundance for only 5 yr. The latter finding exposes an intrinsic danger: sparse lionfish left uncontrolled may derail a concerted, yet not comprehensive, effort to control the invader. This study provides a modelling approach to quantitatively target and manage the lionfish population for nations whose waters are now plagued by invasive lionfish.