We present results from X-ray observations of the quadruply lensed quasar RX J0911.4+0551 using data obtained with the Advanced CCD Imaging Spectrometer ( ACIS ) on board the Chandra X-ray Observatory . The 29 ks observation detects a total of \sim 404 X-ray photons ( 0.3 to 7.0 keV ) from the four images of the lensed quasar . Deconvolution of the aspect corrected data resolves all four lensed images , with relative positions in good agreement with optical measurements . When compared to contemporaneous optical data , one of the lensed images ( component A3 ) is dimmer by a factor of \sim 6 in X-rays with respect to the 2 brighter images ( components A1 and A2 ) . Spectral fitting for the combined images shows significant intrinsic absorption in the soft ( 0.2 to 2.4 keV ) energy band , consistent with the mini-BAL nature of this quasar , while a comparison with ROSAT PSPC observations from 1990 shows a drop of \sim 6.5 in the total soft bandpass flux . The observations also detect \sim 157 X-ray photons arising from extended emission of the nearby cluster ( peaked \sim 42 \arcsec SW of RX J0911.4+0551 ) responsible for the large external shear present in the system . The Chandra observation reveals the cluster emission to be complex and non-spherical , and yields a cluster temperature of kT = 2.3 ^ { +1.8 } _ { -0.8 } keV and a 2.0 to 10 keV cluster luminosity within a 1 Mpc radius of L _ { X } = 7.6 _ { -0.2 } ^ { +0.6 } \times 10 ^ { 43 } ergs s ^ { -1 } ( error bars denote 90 % confidence limits ) . Our mass estimate of the cluster within its virial radius is 2.3 ^ { +1.8 } _ { -0.7 } \times 10 ^ { 14 } \mbox { M } _ { \odot } , and is a factor of two smaller than , although consistent with , previous mass estimates based on the observed cluster velocity dispersion .