Magma flow, heat conduction, and hydrothermal fluid flow control the heat transfer through the upper crust. Magmatic-hydrothermal activity is essential for the utilization of geothermal energy, formation of ore deposits, and predictions of volcanic hazards, but the interplay between magmatic and hydrothermal processes is not well constrained due to the lack of adequate scientific tools. Simulation results from our novel coupled numerical model resolving both magma (Navier-Stokes) and hydrothermal (Darcy) flow quantify the influence of magma convection and rock permeability on energy transfer. Convection of a hot-dry rhyolitic magma has an accelerating effect of up to 15% on the cooling of the reservoir by enhancing the heat transfer toward the magma-hydrothermal interface. However, at high permeabilities and high brittle-ductile transition temperatures, magma flow can also have a secondary decelerating effect, because it prevents efficient permeability creation and entrainment of hydrothermal fluids at the edges of the magma reservoir. Plain Language Summary Magmatic bodies intruded into the Earth's crust at several kilometers depth provide a heat source that drives fluid circulation through permeable rocks. These hydrothermal systems lead to constant degassing of high-temperature fluids in volcanic systems during and between eruption events and can form geothermal resources and ore deposits. At low crystallinities, magma moves within the reservoir. Thus, heat transfer in magmatic-hydrothermal systems occurs by coupled processes of two distinct flow regimes-magma and hydrothermal fluid. Until now, numerical models have focused either on the magmatic or the hydrothermal system. We have developed a fully coupled model that can simulate both processes simultaneously and the interplay between them. We find that rock permeability and thereby the vigor of hydrothermal fluid flow are the primary control on the cooling rates of magmatic intrusions. Magma convection can accelerate this process by up to 15% by efficiently transporting heat toward the magma-hydrothermal interface, where it can be picked up by fluids and transported toward the Earth's surface. However, this process can be counteracted by a secondary deceleration effect that prevents fracturing at high temperatures and thus cooling down of the edges of the magma chamber by hydrothermal fluids.