Deep neural networks (DNNs) have had many successes, but they suffer from two major issues: (1) a vulnerability to adversarial examples and (2) a tendency to elude human interpretation. Interestingly, recent empirical and theoretical evidence suggests these two seemingly disparate issues are actually connected. In particular, robust models tend to provide more interpretable gradients than non-robust models. However, whether this relationship works in the opposite direction remains obscure. With this paper, we seek empirical answers to the following question: can models acquire adversarial robustness when they are trained to have interpretable gradients? We introduce a theoretically inspired technique called Interpretation Regularization (IR), which encourages a model's gradients to (1) match the direction of interpretable target salience maps and (2) have small magnitude. To assess model performance and tease apart factors that contribute to adversarial robustness, we conduct extensive experiments on MNIST and CIFAR-10 with both $\ell_2$ and $\ell_\infty$ attacks. We demonstrate that training the networks to have interpretable gradients improves their robustness to adversarial perturbations. Applying the network interpretation technique SmoothGrad yields additional performance gains, especially in cross-norm attacks and under heavy perturbations. The results indicate that the interpretability of the model gradients is a crucial factor for adversarial robustness. Code for the experiments can be found at https://github.com/a1noack/interp_regularization.