Dust-Obscured Galaxies (DOGs) are bright 24 μm-selected sources with extreme obscuration at optical wavelengths. They are typically characterized by a rising power-law continuum of hot dust (T D ∼ 200–1000 K) in the near-IR indicating that their mid-IR luminosity is dominated by an active galactic nucleus (AGN). DOGs with a fainter 24 μm flux display a stellar bump in the near-IR and their mid-IR luminosity appears to be mainly powered by dusty star formation. Alternatively, it may be that the mid-IR emission arising from AGN activity is dominant but the torus is sufficiently opaque to make the near-IR emission from the AGN negligible with respect to the emission from the host component. In an effort to characterize the astrophysical nature of the processes responsible for the IR emission in DOGs, this paper exploits Herschel data (PACS + SPIRE) on a sample of 95 DOGs within the COSMOS field. We derive a wealth of far-IR properties (e.g. total IR luminosities; mid-to-far-IR colours; dust temperatures and masses) based on spectral energy distribution fitting. Of particular interest are the 24 μm-bright DOGs (F 24 μm > 1 mJy). They present bluer far-IR/mid-IR colours than the rest of the sample, unveiling the potential presence of an AGN. The AGN contribution to the total 8–1000 μm flux increases as a function of the rest-frame 8 μm-luminosity irrespective of the redshift. This confirms that faint DOGs (L 8 μm < 1012 L⊙) are dominated by star formation while brighter DOGs show a larger contribution from an AGN.