We study repeated games where players employ an exponential learning scheme in order to adapt to an ever-changing environment. If the game’s payoffs are subject to random perturbations, this scheme leads to a new stochastic version of the replicator dynamics that is quite different from the “aggregate shocks” approach of evolutionary game theory. Irrespective of the perturbations’ magnitude, we find that strategies which are dominated (even iteratively) eventually become extinct and that the game’s strict Nash equilibria are stochastically asymptotically stable. We complement our analysis by illustrating these results in the case of congestion games.