Madagascar hosts one of the world's most unusual, endemic, diverse and threatened concentrations of fauna. To explain its unique, imbalanced biological diversity, G.G.Simpson proposed the `sweepstakes hypothesis', according to which the ancestors of Madagascar's present-day mammal stock rafted there from Africa. This is an important hypothesis in biogeography and evolutionary theory for how animals colonize new frontiers, but its validity is questioned. Studies suggest that currents were inconsistent with rafting to Madagascar and that land bridges provided the migrants' passage. Here we show that currents could have transported the animals to the island and highlight evidence inconsistent with the land-bridge hypothesis. Using palaeogeographic reconstructions and palaeo-oceanographic modelling, we find that strong surface currents flowed from northeast Mozambique and Tanzania eastward towards Madagascar during the Palaeogene period, exactly as required by the `sweepstakes process'. Subsequently, Madagascar advanced north towards the equatorial gyre and the regional current system evolved into its modern configuration with flows westward from Madagascar to Africa. This may explain why no fully non-aquatic land mammals have colonized Madagascar since the arrival of the rodents and carnivorans during the early-Miocene epoch. One implication is that rafting may be the dominant means of overseas dispersal in the Cenozoic era when palaeocurrent directions are properly considered.