Martian basalts identified by rover in-situ analyses and the study of meteorites represent a direct link to the melting process in the planet's interior and can be used to reconstruct the composition of the mantle and estimate its temperature. Experimentally calibrated numerical models are powerful tools to systematically search for the mantle compositions and melting conditions that can produce melts similar to primary basalts. However, currently available models are not suitable for modeling the melting of FeO-rich peridotites. In this study, we present experiments performed at 1.0 and 2.4–2.6 GPa on a primitive Martian mantle with various P2O5 contents. We use the new experiments together with a comprehensive database of previous melting experiments to calibrate a new model called MAGMARS. This model can reproduce the experimental melt compositions more accurately than Gibbs free energy minimization software (e.g., pMELTS) and can simulate near-fractional polybaric melting of various mantle sources. In addition, we provide an updated thermobarometer that can estimate the P–T melting conditions of primary melts and can be used as a prior step to constrain the input parameters of the MAGMARS melting model. We apply MAGMARS to estimate the source composition of the Adirondack-class basalts and find that melting a depleted mantle, at 2.3–1.7 GPa (Tp = 1390 ± 40°C) can best explain their bulk composition and K2O/Na2O ratio. MAGMARS represents a fast and accurate alternative to calculate the composition of the Martian primary melts and can be used as a stand-alone package or integrated with geodynamical models or other independent modeling software.