This study presents a novel solution to the problem of binaural localization of a speaker in the presence of interfering directional noise and reverberation. Using a state-of-the-art binaural localization algorithm based on a deep neural network (DNN), we propose adding a source separation stage based on non-negative matrix factorization (NMF) to improve the localization performance in conditions with interfering sources. The separation stage is coupled with the localization stage, and is optimized with respect to a broad range of different acoustic conditions, emphasizing a robust and generalizable solution. The machine listening system is shown to greatly benefit from the NMF-based separation stage at low target-to-masker ratios (TMRs) for a variety of noise types, especially for non-stationary noise. It is also demonstrated that training the NMF algorithm on anechoic speech provides better performance than using reverberant speech, and that optimizing the source separation stage using a localization metric rather than a source separation metric substantially increases the system performance.