Understanding the abundances of molecules in dense interstellar clouds requires knowledge of the rates of gas-phase reactions between uncharged species. However, because of the low temperatures within these clouds, reactions with an activation barrier were considered too slow to play an important role. Here we show that, despite the presence of a barrier, the rate coefficient for the reaction between the hydroxyl radical (OH) and methanol--one of the most abundant organic molecules in space--is almost two orders of magnitude larger at 63 K than previously measured at ∼200 K. We also observe the formation of the methoxy radical product, which was recently detected in space. These results are interpreted by the formation of a hydrogen-bonded complex that is sufficiently long-lived to undergo quantum-mechanical tunnelling to form products. We postulate that this tunnelling mechanism for the oxidation of organic molecules by OH is widespread in low-temperature interstellar environments.