We present XMM-Newton observations of three optically-selected z > 0.6 clusters from the ESO Distant Cluster Survey (EDisCS), comprising the first results of a planned X-ray survey of the full EDisCS high-redshift sample. The EDisCS clusters were identified in the Las Campanas Distant Cluster Survey as surface brightness fluctuations in the optical sky and their masses and galaxy populations are well described by extensive photometric and spectroscopic observations. We detect two of the three clusters in the X-ray and place a firm upper limit on diffuse emission in the third cluster field. We are able to constrain the X-ray luminosity and temperature of the detected clusters and estimate their masses. We find the X-ray properties of the detected EDisCS clusters are similar to those of X-ray-selected clusters of comparable mass and -- unlike other high-redshift, optically-selected clusters -- are consistent with the T - sigma and L_x - sigma relations determined from X-ray selected clusters at low redshift. The X-ray determined mass estimates are generally consistent with those derived from weak lensing and spectroscopic analyses. These preliminary results suggest that the novel method of optical selection used to construct the EDisCS catalog may, like selection by X-ray luminosity, be well-suited for identification of relaxed, high-redshift clusters whose intracluster medium is in place and stable by z ~ 0.8.