There is an urgent need to develop novel approaches for predicting and preventing the evolution of antibiotic resistance. Here, we show that the ability to evolve de novo resistance to a clinically important β-lactam antibiotic, ceftazidime, varies drastically across the genus Pseudomonas. This variation arises because strains possessing the ampR global transcriptional regulator evolve resistance at a high rate. This does not arise because of mutations in ampR. Instead, this regulator potentiates evolution by allowing mutations in conserved peptidoglycan biosynthesis genes to induce high levels of β-lactamase expression. Crucially, blocking this evolutionary pathway by co-administering ceftazidime with the β-lactamase inhibitor avibactam can be used to eliminate pathogenic P. aeruginosa populations before they can evolve resistance. In summary, our study shows that identifying potentiator genes that act as evolutionary catalysts can be used to both predict and prevent the evolution of antibiotic resistance.