Given its ubiquity, scale and complexity, few problems have created the combined interest of so many unrelated areas as the evolution of cooperation. Using the tools of evolutionary game theory, here we address, for the first time, the role played by intention recognition in the final outcome of cooperation in large populations of self-regarding individuals. By equipping individuals with the capacity of assessing intentions of others in the course of repeated Prisoner's Dilemma interactions, we show how intention recognition opens a window of opportunity for cooperation to thrive, as it precludes the invasion of pure cooperators by random drift while remaining robust against defective strategies. Intention recognizers are able to assign an intention to the action of their opponents based on an acquired corpus of possible intentions. We show how intention recognizers can prevail against most famous strategies of repeated dilemmas of cooperation, even in the presence of errors. Our approach invites the adoption of other classification and pattern recognition mechanisms common among Humans, to unveil the evolution of complex cognitive processes in the context of social dilemmas.