In the quest to solve the long-standing coronal heating problem, it was suggested half a century ago that coronal loops could be heated by waves. Despite the accumulating observational evidence of the possible importance of coronal waves, still no 3D MHD simulations exist that show significant heating by MHD waves. Here we report on the first 3D coronal loop model that heats the plasma against radiative cooling. The coronal loop is driven at the footpoint by transverse oscillations, and subsequently the induced Kelvin-Helmholtz instability deforms the loop cross section and generates small-scale structures. Wave energy is transferred to smaller scales where it is dissipated, overcoming the internal energy losses by radiation. These results open up a new avenue to address the coronal heating problem.