Despite the structural and functional information contained in the statistical coupling between pairs of residues in a protein, coevolution associated with function is often obscured by artifactual signals such as genetic drift, which shapes a protein's phylogenetic history and gives rise to concurrent variation between protein sequences that is not driven by selection for function. Here, we introduce a background model for phylogenetic contributions of statistical coupling that separates the coevolution signal due to inter-clade and intra-clade sequence comparisons and demonstrate that coevolution can be measured on multiple phylogenetic timescales within a single protein. Our method, nested coevolution (NC), can be applied as an extension to any coevolution metric. We use NC to demonstrate that poorly conserved residues can nonetheless have important roles in protein function. Moreover, NC improved the structural-contact predictions of several coevolution-based methods, particularly in subsampled alignments with fewer sequences. NC also lowered the noise in detecting functional sectors of collectively coevolving residues. Sectors of coevolving residues identified after application of NC were more spatially compact and phylogenetically distinct from the rest of the protein, and strongly enriched for mutations that disrupt protein activity. Thus, our conceptualization of the phylogenetic separation of coevolution provides the potential to further elucidate relationships among protein evolution, function, and genetic diseases.