We present results from a monitoring campaign performed with the Chandra X-ray Observatory of the gravitationally lensed quasars RX J1131-1231 and HE 1104-1805. We detect significant X-ray variability in all images of both quasars. The flux variability detected in image A of RX J1131-1231 is of particular interest because of its high amplitude (a factor of ~ 20). We interpret it as arising from microlensing since the variability is uncorrelated with that of the other images and the X-ray flux ratios show larger changes than the optical as we would expect for microlensing of the more compact X-ray emission regions. The differences between the X-ray and optical flux ratios of HE 1104-1805 are less dramatic, but there is no significant soft X-ray or dust absorption, implying the presence of X-ray microlensing in this system as well. Combining the X-ray data with the optical light curves we find that the X-ray emitting region of HE 1104-1805 is compact with a half-light radius ~ 6 r_g, where the gravitational radius is r_g = 3.6 x 10^14 cm, thus placing significant constraints on AGN corona models. We also find that the microlensing in HE 1104-1805 favors mass models for the lens galaxy that are dominated by dark matter. Finally, we better characterize the massive foreground cluster near RX J1131-1231, set limits on other sources of extended X-ray emission, and limit the fluxes of any central odd images to be 30-50 (3 sigma) times fainter than the observed images. ; Comment: Comments: 23 pages, includes 12 figures, revised version, corrected figure, accepted for publication in ApJ