Protein-DNA and protein-RNA interactions are part of many diverse and essential cellular functions and yet most of them remain to be discovered and characterized. Recent research shows that sequence-based predictors of DNA-binding residues accurately find these residues but also cross-predict many RNA-binding residues as DNA-binding, and vice versa. Most of these methods are also relatively slow, prohibiting applications on the whole-genome scale. We describe a novel sequence-based method, DRNApred, which accurately and in high-throughput predicts and discriminates between DNA- and RNA-binding residues. DRNApred was designed using a new dataset with both DNA- and RNA-binding proteins, regression that penalizes cross-predictions, and a novel two-layered architecture. DRNApred outperforms state-of-the-art predictors of DNA- or RNA-binding residues on a benchmark test dataset by substantially reducing the cross predictions and predicting arguably higher quality false positives that are located nearby the native binding residues. Moreover, it also more accurately predicts the DNA- and RNA-binding proteins. Application on the human proteome confirms that DRNApred reduces the cross predictions among the native nucleic acid binders. Also, novel putative DNA/RNA-binding proteins that it predicts share similar subcellular locations and residue charge profiles with the known native binding proteins. Webserver of DRNApred is freely available at http://biomine.cs.vcu.edu/servers/DRNApred/.