Deep-marine volcanism drives Earth's most energetic transfers of heat and mass between the crust and the oceans. While magmatic activity on the seafloor has been correlated with the occurrence of colossal enigmatic plumes of hydrothermal fluid known as megaplumes, little is known of the primary source and intensity of the energy release associated with seafloor volcanism. As a result, the specific origin of megaplumes remains ambiguous. By developing a mathematical model for the dispersal of submarine tephras, we show that the transport of pyroclasts requires an energy discharge that is sufficiently powerful (~1-2 TW) to form a hydrothermal plume with characteristics matching those of observed megaplumes in a matter of hours. Our results thereby directly link megaplume creation, active magma extrusion, and tephra dispersal. The energy flux at the plume source required to drive the dispersal is difficult to attain by purely volcanogenic means, and likely requires an additional input of heat, potentially from rapid evacuations of hot hydrothermal fluids triggered by dyke intrusion. In view of the ubiquity of submarine tephra deposits, our results demonstrate that intervals of rapid hydrothermal discharge are likely commonplace during deep-ocean volcanism.