We present the X-ray properties of 108 Dust-Obscured Galaxies (DOGs; F24 m/FR > 1000) in the COSMOS field, all of which are detected in at least three far-infrared bands with the Herschel Observatory. Out of the entire sample, 22 are individually detected in the hard 2-8 keV X-ray band by the Chandra COSMOS Legacy survey, allowing us to classify them as AGN. Six (27%) of them are Compton-thick AGN candidates with column densities NH > 1024 cm−2, while 15 are moderately obscured AGNs with 1022 < NH < 1024 cm−2. Additionally, we estimate AGN contributions to the IR luminosity (8-1000 m rest-frame) greater than 20% for 19 DOGs based on SED decomposition using Spitzer/MIPS 24 m and the five Herschel bands (100-500 m). Only 7 of these are detected in X-rays individually. We performed an X-ray stacking analysis for the 86 undetected DOGs. We find that the AGN fraction in DOGs increases with 24 m flux and that it is higher than that of the general 24 m population. However, no significant difference is found when considering only X-ray detections. This strongly motivates the combined use of X-ray and far-IR surveys to successfully probe a wider population of AGNs, particularly for the most obscured ones.