We present the first pointed X-ray observation of DW Ursae Majoris, a novalike cataclysmic variable (CV) and one of the archetype members of the SW Sextantis class, obtained with the XMM-Newton satellite. These data provide the first detailed look at an SW Sex star in the X-ray regime (with previous X-ray knowledge of the SW Sex stars limited primarily to weak or non-detections in the ROSAT All Sky Survey). It is also one of only a few XMM-Newton observations (to date) of any high mass transfer rate novalike CV, and the only one in the evolutionarily important 3-4 hr orbital period range. The observed X-ray spectrum of DW UMa is very soft, with ~95% of the detected X-ray photons at energies <2 keV. The spectrum can be fit equally well by a one-component cooling flow model, with a temperature range of 0.2-3.5 keV, or a two-component, two-temperature thermal plasma model, containing hard (~5-6 keV) and soft (~0.8 keV) components. The X-ray light curve of DW UMa shows a likely partial eclipse, implying X-ray reprocessing in a vertically extended region, and an orbital modulation, implying a structural asymmetry in the X-ray reprocessing site (e.g., it cannot be a uniform corona). We also obtained a simultaneous near-ultraviolet light curve of DW UMa using the Optical Monitor on XMM-Newton. This light curve is similar in appearance to published optical-UV light curves of DW UMa and shows a prominent deep eclipse. Regardless of the exact nature of the X-ray reprocessing site in DW UMa, the lack of a prominent hard X-ray total eclipse and very low fraction of high energy X-rays point to the presence of an optically and geometrically thick accretion disk that obscures the boundary layer and modifies the X-ray spectrum emitted near the white dwarf.