We have analyzed Einstein High Resolution Imager (HRI) and Imaging Proportional Counter (IPC) observations of the gravitationally lensed quasar 0957 + 561. The X-ray emission appears as an approximate semicircle extending from the northern quasar image (A) to the southern image (B). We find unresolved X-ray emission associated with B, while A lies in an extended region. Although, for the epoch of the X-ray observations, the ratio of the optical flux from image B to that from image A is 0.76, the HRI observations show that the emission from around quasar image B is only about one-third that from image A. This difference in ratios implies the presence of another component in 0957 + 561. Since this ratio is found in each of the two HRI observations which were taken 6 months apart, it is unlikely that the difference in the ratios observed in X-ray and optical wavelengths is caused by dramatic changes in the quasar flux on a time scale less than the difference in the light propagation time for the two images. We use a model for the lens, including a giant galaxy and its surrounding cluster, to infer the X-ray source distribution and luminosity. This model implies the existence of an extended source of X-ray emission offset from the unresolved source associated with the quasar nucleus. The extended component has a diameter approximately 1'' to 3'', offset 0.''7 northeast of the quasar. The IPC observation of 0957 + 561 shows that the spectrum of the emission is soft compared to that of other X-ray emitting quasars. We discuss possible sources for the origin of the X-ray emission and suggest that it could arise from radiative cooling of hot gas surrounding the quasar. We also suggest that the detection of the extended emission in 0957 + 561 is possible only because of the large magnification provided by the intervening lens.