Recent progress in the study of resonant light confinement in high-index dielectric nanostructures suggests a new route for achieving efficient control of both electric and magnetic components of light. It also leads to the enhancement of nonlinear effects near electric and magnetic Mie resonances with an engineered radiation directionality. Here we study the third-harmonic generation from dimers composed of pairs of two identical silicon nanoparticles and demonstrate, both numerically and experimentally, that the multipolar harmonic modes generated by the dimers near the Mie resonances allow the shaping of the directionality of nonlinear radiation.