High harmonic generation (HHG) in monolayer MoS2 is studied using fully microscopic many-body models based on the semiconductor Bloch equations and density functional theory. It is shown that Coulomb correlations lead to a dramatic enhancement of HHG. In particular, near the bandgap, enhancements of two orders of magnitude or more are observed for a wide range of excitation wavelengths and intensities. For excitation at excitonic resonances, strong absorption leads to spectrally broad sub-floors of the harmonics that is absent without Coulomb interaction. The widths of these sub-floors depend strongly on the dephasing time for polarizations. For times of the order of 10 fs the broadenings are comparable to the Rabi energies and reach one electronvolt at fields of approximately 50 MV/cm. The intensities of these contributions are approximately four to six orders below the peaks of the harmonics.