Robustness and compactness are two essential attributes of deep learning models that are deployed in the real world. The goals of robustness and compactness may seem to be at odds, since robustness requires generalization across domains, while the process of compression exploits specificity in one domain. We introduce \textit{Adaptive Sharpness-Aware Pruning (AdaSAP)}, which unifies these goals through the lens of network sharpness. The AdaSAP method produces sparse networks that are robust to input variations which are \textit{unseen at training time}. We achieve this by strategically incorporating weight perturbations in order to optimize the loss landscape. This allows the model to be both primed for pruning and regularized for improved robustness. AdaSAP improves the robust accuracy of pruned models on classification and detection over recent methods by up to +6\% on OOD datasets, over a wide range of compression ratios, pruning criteria, and architectures.