Understanding how novel functions evolve (genetic adaptation) is a critical goal of evolutionary biology. Among asexual organisms, genetic adaptation involves multiple mutations that frequently interact in a non-linear fashion (epistasis). Non-linear interactions pose a formidable challenge for the computational prediction of mutation effects. Here we use the recent evolution of β-lactamase under antibiotic selection as a model for genetic adaptation. We build a network of coevolving residues (possible functional interactions), in which nodes are mutant residue positions and links represent two positions found mutated together in the same sequence. Most often these pairs occur in the setting of more complex mutants. Focusing on extended-spectrum resistant sequences, we use network-theoretical tools to identify triple mutant trajectories of likely special significance for adaptation. We extrapolate evolutionary paths (n = 3) that increase resistance and that are longer than the units used to build the network (n = 2). These paths consist of a limited number of residue positions and are enriched for known triple mutant combinations that increase cefotaxime resistance. We find that the pairs of residues used to build the network frequently decrease resistance compared to their corresponding singlets. This is a surprising result, given that their coevolution suggests a selective advantage. Thus, β-lactamase adaptation is highly epistatic. Our method can identify triplets that increase resistance despite the underlying rugged fitness landscape and has the unique ability to make predictions by placing each mutant residue position in its functional context. Our approach requires only sequence information, sufficient genetic diversity, and discrete selective pressures. Thus, it can be used to analyze recent evolutionary events, where coevolution analysis methods that use phylogeny or statistical coupling are not possible. Improving our ability to assess evolutionary trajectories will help predict the evolution of clinically relevant genes and aid in protein design.