We present results of X-ray observations of a sample of 15 clusters selected via their imprint on the cosmic microwave background from the thermal Sunyaev-Zel'dovich (SZ) effect. These clusters are a subset of the first SZ-selected cluster catalog, obtained from observations of 178 deg(2 )of sky surveyed by the South Pole Telescope (SPT). Using X-ray observations with Chandra and XMM-Newton, we estimate the temperature, T(X) , and mass, M(g) , of the intracluster medium within r (500) for each cluster. From these, we calculate Y(X) = M(g)T(X) and estimate the total cluster mass using an M (500-)Y(X) scaling relation measured from previous X-ray studies. The integrated Comptonization, Y (SZ), is derived from the SZ measurements, using additional information from the X-ray-measured gas density profiles and a universal temperature profile. We calculate scaling relations between the X-ray and SZ observables and find results generally consistent with other measurements and the expectations from simple self-similar behavior. Specifically, we fit a Y (SZ-)Y(X) relation and find a normalization of 0.82 ± 0.07, marginally consistent with the predicted ratio of Y (SZ)/Y(X) = 0.91 ± 0.01 that would be expected from the density and temperature models used in this work. Using the Y(X -)derived mass estimates, we fit a Y (SZ-)M (500) relation and find a slope consistent with the self-similar expectation of Y (SZ) M (5/3) with a normalization consistent with predictions from other X-ray studies. We find that the SZ mass estimates, derived from cosmological simulations of the SPT survey, are lower by a factor of 0.78 ± 0.06 relative to the X-ray mass estimates. This offset is at a level of 1.3σ when considering the ~15% systematic uncertainty for the simulation-based SZ masses. Overall, the X-ray measurements confirm that the scaling relations of the SZ-selected clusters are consistent with the properties of other X-ray-selected samples of massive clusters, even allowing for the broad redshift range (0.29