Mn₅Si₃ displays an unusual and complex magnetic ground state, which is considered to be the origin of the anomalous transport and thermodynamic properties that it exhibits. We report the magnetic exchange couplings of the noncollinear antiferromagnetic phase of Mn₅Si₃ using inelastic neutron scattering measurements and density functional theory calculations. We determine the ground-state spin configuration and compute its magnon dispersion relations which are in good agreement with the ones obtained experimentally. Furthermore, we investigate the evolution of the spin texture under the application of an external magnetic field to demonstrate theoretically the multiple field-induced phase transitions that Mn₅Si₃ undergoes. Finally, we model the stability of some of the material’s magnetic moments under a magnetic field and we find that very susceptible magnetic moments in a frustrated arrangement can be tuned by the field. This data set contains the data relevant to perform the DFT calculations with the JuKKR code, the spin dynamics simulations with the Spirit code, the spin-wave calculations with the SWIS code, and the neutron scattering experimental data.