Evolution by natural selection tends to favors those that replicate faster to leave more offspring; nature, however, abounds with examples where organisms seem to pay a reproductive cost to disperse faster. When does selection favor this ‘survival of the fastest?’ We searched for a simple rule, motivated by evolution experiments where swarming bacteria evolved into an hyperswarmer mutant which disperses ~100% faster but pays a growth cost of ~10 % to make many copies of its flagellum. We analyzed a two-species model based on the Fisher equation to explain this observation: the rate of swarming expansion (v) results from an interplay of growth (r) and dispersal (D) and is independent of the carrying capacity: v=2sqrt(rD). A mutant can take over the edge only if its expansion rate (v_2) exceeds the expansion rate of the established species’ (v_1), and this simple condition (v_2>v_1) determines the maximum cost in slower growth that a hyperswarmer can pay and still be able to take over. Numerical simulations and time-course experiments where we tracked evolution by imaging bacteria suggest that our findings are general: less favorable conditions delay but do not entirely prevent the success of the fastest. The same principles should apply to other expanding populations where we observe the selection for faster dispersal despite a cost to growth, including territorial occupation by non-native species and the spread of cancer metastasis.