Machine learning from explanations (MLX) is an approach to learning that uses human-provided annotations of relevant features for each input to ensure that model predictions are right for the right reasons. Existing MLX approaches rely heavily on a specific model interpretation approach and require strong parameter regularization to align model and human explanations, leading to sub-optimal performance. We recast MLX as an adversarial robustness problem, where human explanations specify a lower dimensional manifold from which perturbations can be drawn, and show both theoretically and empirically how this approach alleviates the need for strong parameter regularization. We consider various approaches to achieving robustness, leading to improved performance over prior MLX methods. Finally, we combine robustness with an earlier MLX method, yielding state-of-the-art results on both synthetic and real-world benchmarks.