Dissipation and the accompanying fluctuations are often seen as detrimental for quantum systems since they are associated with fast relaxation and loss of phase coherence. However, it has been proposed that a pure state can be prepared if external noise induces suitable downwards transitions, while exciting transitions are blocked. We demonstrate such a refrigeration mechanism in a cavity optomechanical system, where we prepare a mechanical oscillator in its ground state by injecting strong electromagnetic noise at frequencies around the red mechanical sideband of the cavity. The optimum cooling is reached with a noise bandwidth smaller than but on the order of the cavity decay rate. At higher bandwidths, cooling is less efficient as suitable transitions are not effectively activated. In the opposite regime where the noise bandwidth becomes comparable to the mechanical damping rate, damping follows the noise amplitude adiabatically, and the cooling is also suppressed.