Surveying with multibeam echo sounders around old (≫1 Ma) volcanic ocean islands reveals that their submarine flanks contain a strong downslope-oriented ridge-and-valley corrugation, which modifies the original volcanic morphology of lava terraces and cones. By analogy with canyons in other settings, this corrugation was probably caused by channel incision by erosive sedimentary mass flows such as turbidity currents and debris flows. We adapt a method first used in subaerial geomorphology to isolate the erosion depth (exhumation) and apply it to the eroded flanks of the 6--8 Ma Anaga massif of Tenerife. The channels formed around this massif divert around local topographic highs. These highs, which are probably original volcanic cones, are therefore preferentially preserved during erosion, so that their elevations can be used to construct an artificial reference surface. Terrain depth was calculated by subtracting this reference surface from measured bathymetry. Comparison of the terrain depth of the old, eroded submarine flank of Anaga with that of the young, mostly unaltered submarine flank of El Hierro allows us to infer the mean depth of Anaga's submarine erosion, which is ~100 m. Volcanic terrains can be dated by radiometric methods, so they also provide a way of quantifying long-term denudation rates. We infer that submarine denudation of Anaga has occurred at comparable rates to that of subaerial lowlands and much slower than denudation of highlands, illustrated locally by the more extensive erosion of the subaerial Anaga edifice.