Spatial structure is one of the most studied mechanisms in evolutionary game theory. Here, we explore the consequences of spatial structure for a question which has received considerable empirical and theoretical attention in recent years, but has not yet been studied from a network perspective: whether cooperation relies on intuitive predispositions or deliberative self-control. We examine this question using a model which integrates the “dual-process” framework from cognitive science with evolutionary game theory, and considers the evolution of agents who are embedded within a social network and only interact with their neighbors. In line with past work in well-mixed populations, we find that selection favors either the intuitive defector (ID) strategy which never deliberates, or the dual-process cooperator (DC) strategy which intuitively cooperates but uses deliberation to switch to defection in Prisoner’s Dilemma games. We find that sparser networks (i.e. smaller average degree) facilitate the success of DC over ID, while also reducing the level of deliberation that DC agents engage in; and that these results generalize across different kinds of networks. These observations demonstrate the important role that spatial structure can have not just on the evolution of cooperation, but on the co-evolution of cognition and cooperation.