Biological invasions are one of the major threats to both ecosystem and economic functioning. Their management typically involves culling of the pest or removal of its habitat. The Asiatic red-bellied beautiful squirrel Callosciurus erythraeus is the first known introduction of a squirrel into South America. It established from five releases in 1973, using exotic trees to spread through Argentinean Pampas. It now causes substantial economic damage in agricultural and urban areas across >680 km2, and its continued spread threatens indigenous species. We developed a spatially explicit model of the invasion for the likely range of life-history parameters, matched against empirical data on patch occupancy in 2004. The two best-fitting models suggest the current population to be on the cusp of an explosive expansion. These models were used to predict future trends under alternative scenarios of strategic culling or habitat removal aimed at slowing the spread. The predictions for 18 yr into the future were that 1) the present lack of systematic management will lead to a 5-fold increase in area of occupancy, 2) removal of habitat down to half carrying capacity will thin the population but accelerate its spread, 3) 10 yr of culling above the maximum sustainable yield (MSY) will precipitate declines in abundance and patch occupancy towards extinction, but with immediate recovery upon cessation of the cull. We recommend continuous culling above the MSY in priority patches, aimed at slowing arrival to valuable conservation areas. This study demonstrates the need for prompt action to terminate invasions before they establish. The squirrel invasion is now irreversible after 30 yr of slow spread across fragmented habitat. Although culling requires public awareness campaigns and sustained governmental commitment, it is now the best feasible strategy for managing this invasion.