Simplicity bias is the concerning tendency of deep networks to over-depend on simple, weakly predictive features, to the exclusion of stronger, more complex features. This causes biased, incorrect model predictions in many real-world applications when trained on incomplete data containing spurious feature-label correlations. We propose a direct, interventional method for addressing simplicity bias in DNNs, called the feature sieve. We aim to automatically identify and suppress easily-computable features in lower layers of the network, thereby allowing the higher network levels to extract and utilize richer, more meaningful representations. We provide concrete evidence of this differential suppression & enhancement of features using controlled datasets, and report substantial gains on many real-world debiasing benchmarks (11.4\% relative gain on ImageNet-A; 3.2% on BAR, etc). Crucially, we outperform many baselines that incorporate knowledge about ``simple'' features, or known spurious attributes, despite our method not using any such information. We believe that our feature sieve work opens up exciting new research directions in automatic adversarial feature extraction techniques for deep networks.