In this work we proposed to constrain sub-GeV dark matter particles (1 MeV $\leq$ m$_\textrm{DM}$ $\leq$ 5 GeV) by looking at energies much lower than the mass range of interest, by means of the inverse Compton scattering contribution to the total flux. In particular, the electrons and positrons produced by dark matter particles give rise to X rays by up-scattering the low-energy photons in the Milky Way, notably cosmic microwave background, infrared light from dust and optical starlight. These X rays fall in the energy range covered by the {\sc Integral} data, which we used to constrain the dark matter annihilation cross-section. We considered three annihilation channels: electron, muon and pion. As a result, we derived competitive constraints for dark matter particles with a mass between 150 MeV and 1.5 GeV.