The Iberian lynx (Lynx pardinus) has suffered severe population declines in the twentieth century and is now on the brink of extinction. Climate change could further threaten the survival of the species, but its forecast effects are being neglected in recovery plans. Quantitative estimates of extinction risk under climate change have so far mostly relied on inferences from correlative projections of species' habitat shifts. Here we use ecological niche models coupled to metapopulation simulations with source-sink dynamics to directly investigate the combined effects of climate change, prey availability and management intervention on the persistence of the Iberian lynx. Our approach is unique in that it explicitly models dynamic bi-trophic species interactions in a climate change setting. We show that anticipated climate change will rapidly and severely decrease lynx abundance and probably lead to its extinction in the wild within 50 years, even with strong global efforts to mitigate greenhouse gas emissions. In stark contrast, we also show that a carefully planned reintroduction programme, accounting for the effects of climate change, prey abundance and habitat connectivity, could avert extinction of the lynx this century. Our results demonstrate, for the first time, why considering prey availability, climate change and their interaction in models is important when designing policies to prevent future biodiversity loss.