Inverse problems aim to reconstruct unseen data from corrupted or perturbed measurements. While most work focuses on improving reconstruction quality, generalization accuracy and robustness are equally important, especially for safety-critical applications. Model-based architectures (MBAs), such as loop unrolling methods, are considered more interpretable and achieve better reconstructions. Empirical evidence suggests that MBAs are more robust to perturbations than black-box solvers, but the accuracy-robustness tradeoff in MBAs remains underexplored. In this work, we propose a simple yet effective training scheme for MBAs, called SGD jittering, which injects noise iteration-wise during reconstruction. We theoretically demonstrate that SGD jittering not only generalizes better than the standard mean squared error training but is also more robust to average-case attacks. We validate SGD jittering using denoising toy examples, seismic deconvolution, and single-coil MRI reconstruction. Both SGD jittering and its SPGD extension yield cleaner reconstructions for out-of-distribution data and demonstrates enhanced robustness against adversarial attacks.