Sequence-based neural networks can learn to make accurate predictions from large biological datasets, but model interpretation remains challenging. Many existing feature attribution methods are optimized for continuous rather than discrete input patterns and assess individual feature importance in isolation, making them ill-suited for interpreting non-linear interactions in molecular sequences. Building on work in computer vision and natural language processing, we developed an approach based on deep generative modeling - Scrambler networks - wherein the most salient sequence positions are identified with learned input masks. Scramblers learn to generate Position-Specific Scoring Matrices (PSSMs) where unimportant nucleotides or residues are 'scrambled' by raising their entropy. We apply Scramblers to interpret the effects of genetic variants, uncover non-linear interactions between cis-regulatory elements, explain binding specificity for protein-protein interactions, and identify structural determinants of de novo designed proteins. We show that interpretation based on a generative model allows for efficient attribution across large datasets and results in high-quality explanations, often outperforming state-of-the-art methods.