We compare the surface brightness profile and morphology of the Galactic Centre Excess (GCE) identified in wide-angle γ-ray maps from the Fermi-Large Area Telescope (LAT) to dark matter annihilation predictions derived from high-resolution Λ cold dark matter magnetohydrodynamic simulations of galaxy formation. These simulations produce isolated, disc-dominated galaxies with structure, stellar populations, gas content, and stellar and halo masses comparable to those of the Milky Way. For a specific choice of annihilation cross-section, they agree well with the Fermi-LAT data over the full observed angular range, 1°-15°, whereas their dark-matter-only counterparts, lacking any compression of the inner halo by the gravitational effects of the baryons, fail to predict emission as centrally concentrated as observed. These results provide additional support to the hypothesis that the GCE is produced by annihilating dark matter. If, however, it is produced by a different mechanism, they imply a strong upper limit on annihilation rates, which can be translated into upper limits on the expected γ-ray flux not only from the inner Galaxy, but also from any substructure, with or without stars, in the Galactic halo.