Specific protein-protein interactions are crucial in the cell, both to ensure the formation and stability of multi-protein complexes, and to enable signal transduction in various pathways. Functional interactions between proteins result in coevolution between the interaction partners, causing their sequences to be correlated. Here we exploit these correlations to accurately identify which proteins are specific interaction partners from sequence data alone. Our general approach, which employs a pairwise maximum entropy model to infer couplings between residues, has been successfully used to predict the three-dimensional structures of proteins from sequences. Thus inspired, we introduce an iterative algorithm to predict specific interaction partners from two protein families whose members are known to interact. We first assess the algorithms performance on histidine kinases and response regulators from bacterial two-component signaling systems. We obtain a striking 0.93 true positive fraction on our complete dataset without any a priori knowledge of interaction partners, and we uncover the origin of this success. We then apply the algorithm to proteins from ATP-binding cassette (ABC) transporter complexes, and obtain accurate predictions in these systems as well. Finally, we present two metrics that accurately distinguish interacting protein families from non-interacting ones, using only sequence data.