Brain-inspired learning in physical hardware has enormous potential to learn fast at minimal energy expenditure. One of the characteristics of biological learning systems is their ability to learn in the presence of various noise sources. Inspired by this observation, we introduce a novel noise-based learning approach for physical systems implementing multi-layer neural networks. Simulation results show that our approach allows for effective learning whose performance approaches that of the conventional effective yet energy-costly backpropagation algorithm. Using a spintronics hardware implementation, we demonstrate experimentally that learning can be achieved in a small network composed of physical stochastic magnetic tunnel junctions. These results provide a path towards efficient learning in general physical systems which embraces rather than mitigates the noise inherent in physical devices.