Data poisoning and backdoor attacks manipulate training data to induce security breaches in a victim model. These attacks can be provably deflected using differentially private (DP) training methods, although this comes with a sharp decrease in model performance. The InstaHide method has recently been proposed as an alternative to DP training that leverages supposed privacy properties of the mixup augmentation, although without rigorous guarantees. In this work, we show that strong data augmentations, such as mixup and random additive noise, nullify poison attacks while enduring only a small accuracy trade-off. To explain these finding, we propose a training method, DP-InstaHide, which combines the mixup regularizer with additive noise. A rigorous analysis of DP-InstaHide shows that mixup does indeed have privacy advantages, and that training with k-way mixup provably yields at least k times stronger DP guarantees than a naive DP mechanism. Because mixup (as opposed to noise) is beneficial to model performance, DP-InstaHide provides a mechanism for achieving stronger empirical performance against poisoning attacks than other known DP methods.