Selection for more efficient socially learned behaviors over alternatives is crucial for cumulative cultural evolution, yet our understanding of such cultural selection in animals is limited. We performed a cultural diffusion experiment using 18 populations of wild-caught great tits Parus major to ask whether more efficient foraging traditions are selected for, and whether this process is affected by turnover. We show that gradual replacement of individuals greatly increases the probability that a more efficient behavior will invade a population's cultural repertoire, out-competing an established inefficient behavior. Turnover does not increase innovation rates, but instead increases adoption rates, as immigrants are more susceptible to novel, efficient behaviors. An agent based model further supported our results by demonstrating that this effect holds across populations of different types of learners. Altogether, these results provide strong evidence for cultural selection for efficiency in animals, and highlight the importance of population turnover for this process.