We present results from monitoring observations of the gravitationally lensed quasar RX J1131-1231 performed with the Chandra X-ray Observatory. The X-ray observations were planned with relatively long exposures that allowed a search for energy-dependent microlensing in the soft (0.2-2 keV) and hard (2-10 keV) light curves of the images of RX J1131-1231. We detect significant microlensing in the X-ray light-curves of images A and D, and energy-dependent microlensing of image D. The magnification of the soft band appears to be larger than that in the hard band by a factor of ~ 1.3 when image D becomes more magnified. This can be explained by the difference between a compact, softer-spectrum corona that is producing a more extended, harder spectrum reflection component off the disk. This is supported by the evolution of the fluorescent iron line in image D over three consecutive time-averaged phases of the light curve. In the first period, a Fe line at E = 6.36(-0.16,+0.13) keV is detected (at > 99% confidence). In the second period, two Fe lines are detected, one at E = 5.47(-0.08,+0.06) keV (detected at > 99% confidence) and another at E = 6.02(-0.07,+0.09) keV (marginally detected at > 90% confidence), and in the third period, a broadened Fe line at 6.42(-0.15,+0.19) keV is detected (at > 99% confidence). This evolution of the Fe line profile during the microlensing event is consistent with the line distortion expected when a caustic passes over the inner disk where the shape of the fluorescent Fe line is distorted by General Relativistic and Doppler effects. ; Comment: 20 pages, includes 10 figures, submitted to ApJ