We study the evolution of cooperation in the framework of evolutionary game theory, adopting the prisoner's dilemma and snowdrift game as metaphors of cooperation between unrelated individuals. In sharp contrast with previous results we find that, whenever individuals interact following networks of contacts generated via growth and preferential attachment, leading to strong correlations between individuals, cooperation becomes the dominating trait throughout the entire range of parameters of both games, as such providing a unifying framework for the emergence of cooperation. Such emergence is shown to be inhibited whenever the correlations between individuals are decreased or removed. These results are shown to apply from very large population sizes down to small communities with nearly 100 individuals.