Recent model inversion attack algorithms permit adversaries to reconstruct a neural network's private training data just by repeatedly querying the network and inspecting its outputs. In this work, we develop a novel network architecture that leverages sparse-coding layers to obtain superior robustness to this class of attacks. Three decades of computer science research has studied sparse coding in the context of image denoising, object recognition, and adversarial misclassification settings, but to the best of our knowledge, its connection to state-of-the-art privacy vulnerabilities remains unstudied. However, sparse coding architectures suggest an advantageous means to defend against model inversion attacks because they allow us to control the amount of irrelevant private information encoded in a network's intermediate representations in a manner that can be computed efficiently during training and that is known to have little effect on classification accuracy. Specifically, compared to networks trained with a variety of state-of-the-art defenses, our sparse-coding architectures maintain comparable or higher classification accuracy while degrading state-of-the-art training data reconstructions by factors of 1.1 to 18.3 across a variety of reconstruction quality metrics (PSNR, SSIM, FID). This performance advantage holds across 5 datasets ranging from CelebA faces to medical images and MNIST, and across various state-of-the-art SGD-based and GAN-based inversion attacks, including Plug-and-Play attacks. We provide a comprehensive performance evaluation codebase to encourage further research in this regard.