Determining which proteins interact together is crucial to a systems-level understanding of the cell. Recently, algorithms based on Direct Coupling Analysis (DCA) pairwise maximum-entropy models have allowed to identify interaction partners among paralogous proteins from sequence data. This success of DCA at predicting protein-protein interactions could be mainly based on its known ability to identify pairs of residues that are in contact in the three-dimensional structure of protein complexes and that coevolve to remain physicochemically complementary. However, interacting proteins possess similar evolutionary histories. What is the role of purely phylogenetic correlations in the performance of DCA-based methods to infer interaction partners? To address this question, we employ controlled synthetic data that only involve phylogeny and no interactions or contacts. We find that DCA accurately identifies the pairs of synthetic sequences that share evolutionary history. While phylogenetic correlations confound the identification of contacting residues by DCA, they are thus useful to predict interacting partners among paralogs. We find that DCA performs as well as phylogenetic methods to this end, and slightly better than them with large and accurate training sets. Employing DCA or phylogenetic methods within an Iterative Pairing Algorithm (IPA) allows to predict pairs of evolutionary partners without a training set. We further demonstrate the ability of these various methods to correctly predict pairings among real paralogous proteins with genome proximity but no known direct physical interaction, illustrating the importance of phylogenetic correlations in natural data. However, for physically interacting and strongly coevolving proteins, DCA and mutual information outperform phylogenetic methods. We finally discuss how to distinguish physically interacting proteins from proteins that only share a common evolutionary history.