Enhancement of optical response with high-index dielectric nanoparticles is attributed to the excitation of their Mie-type magnetic and electric resonances. Here we study Raman scattering from crystalline silicon nanoparticles and reveal that magnetic dipole modes have a much stronger effect on the scattering than electric modes of the same order. We demonstrate experimentally a 140-fold enhancement of the Raman signal from individual silicon spherical nanoparticles at the magnetic dipole resonance. Our results confirm the importance of the optically-induced magnetic response of subwavelength dielectric nanoparticles for enhancing light-matter interactions.