Continental rift systems are commonly characterized by volcanism with parental basaltic magmas sourced from the mantle. Erosion of the rift shoulders and sedimentation in the adjacent basins can affect the stress and thermal fields at depth, thereby affecting partial mantle melting. However, the sensitivity of magmatic activity to such surface forcing is elusive. Geological observations from the western Ross Sea, Antarctica, suggest rift onset in the Cretaceous with a transition from wide-rifting to narrow-rifting at the boundary between the Antarctic craton and the Transantarctic Mountains. Miocene climate cooling during rifting in the western Ross Sea, in addition, leads to an abrupt decrease in sedimentation rate, synchronous to the emplacement of the McMurdo Volcanic Group. This represents the largest alkali province worldwide, extending both inland and offshore of Transantarctic Mountains and western Ross Sea, respectively. Here, we use coupled thermo-mechanical and landscape evolution numerical modeling to quantify melt production in slowly stretching rift basins due to changes in erosion/deposition rates. The model combines visco-elasto-plastic deformation of the lithosphere and underlying mantle during extension, partial rock melting, and linear hillslope diffusion of the surface topography. The parametric study covers a range of slow extension rates, crustal thicknesses, mantle potential temperatures and diffusion coefficients. Numerical simulations successfully reproduce the ∼150–200-km-wide extension of western Ross Sea and Miocene-to-present asthenospheric melt production (McMurdo Volcanic Group). Results further show that slow rifts magmatism is highly sensitive to sediment deposition within the basin, which inhibits mantle decompression melting and delays the crustal breakup. Regional climate-driven sedimentation rate changes are thus likely to have affected the syn-rift magmatic history of the western Ross Sea, Antarctica, supporting the relevance of interactions between surface and deep-seated processes across extensional settings.