Hebbian plasticity in winner-take-all (WTA) networks is highly attractive for neuromorphic on-chip learning, owing to its efficient, local, unsupervised, and on-line nature. Moreover, its biological plausibility may help overcome important limitations of artificial algorithms, such as their susceptibility to adversarial attacks, and their high demands for training-example quantity and repetition. However, Hebbian WTA learning has found little use in machine learning (ML), likely because it has been missing an optimization theory compatible with deep learning (DL). Here we show rigorously that WTA networks constructed by standard DL elements, combined with a Hebbian-like plasticity that we derive, maintain a Bayesian generative model of the data. Importantly, without any supervision, our algorithm, SoftHebb, minimizes cross-entropy, i.e. a common loss function in supervised DL. We show this theoretically and in practice. The key is a "soft" WTA where there is no absolute "hard" winner neuron. Strikingly, in shallow-network comparisons with backpropagation (BP), SoftHebb shows advantages beyond its Hebbian efficiency. Namely, it converges in fewer iterations, and is significantly more robust to noise and adversarial attacks. Notably, attacks that maximally confuse SoftHebb are also confusing to the human eye, potentially linking human perceptual robustness, with Hebbian WTA circuits of cortex. Finally, SoftHebb can generate synthetic objects as interpolations of real object classes. All in all, Hebbian efficiency, theoretical underpinning, cross-entropy-minimization, and surprising empirical advantages, suggest that SoftHebb may inspire highly neuromorphic and radically different, but practical and advantageous learning algorithms and hardware accelerators.