Robustness and compactness are two essential components of deep learning models that are deployed in the real world. The seemingly conflicting aims of (i) generalization across domains as in robustness, and (ii) specificity to one domain as in compression, are why the overall design goal of achieving robust compact models, despite being highly important, is still a challenging open problem. We introduce Adaptive Sharpness-Aware Pruning, or AdaSAP, a method that yields robust sparse networks. The central tenet of our approach is to optimize the loss landscape so that the model is primed for pruning via adaptive weight perturbation, and is also consistently regularized toward flatter regions for improved robustness. This unifies both goals through the lens of network sharpness. AdaSAP achieves strong performance in a comprehensive set of experiments. For classification on ImageNet and object detection on Pascal VOC datasets, AdaSAP improves the robust accuracy of pruned models by +6% on ImageNet C, +4% on ImageNet V2, and +4% on corrupted VOC datasets, over a wide range of compression ratios, saliency criteria, and network architectures, outperforming recent pruning art by large margins.