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The robustness and anomaly detection capability of neural networks are crucial topics for their safe adoption in the real-world. Moreover, the over-parameterization of recent networks comes with high computational costs and raises questions about its influence on robustness and anomaly detection. In this work, we show that sparsity can make networks more robust and better anomaly detectors. To motivate this even further, we show that a pre-trained neural network contains, within its parameter space, sparse subnetworks that are better at these tasks without any further training. We also show that structured sparsity greatly helps in reducing the complexity of expensive robustness and detection methods, while maintaining or even improving their results on these tasks. Finally, we introduce a new method, SensNorm, which uses the sensitivity of weights derived from an appropriate pruning method to detect anomalous samples in the input.