As part of a multiwavelength study of the unusual radio supernova remnant DA 495, we present observations made with the Chandra X-ray Observatory. Imaging and spectroscopic analysis confirms the previously detected X-ray source at the heart of the annular radio nebula, establishing the radiative properties of two key emission components: a soft unresolved source with a blackbody temperature of 1 MK consistent with a neutron star, surrounded by a nonthermal nebula 40 in diameter exhibiting a power-law spectrum with photon index Gamma = 1.6+/-0.3, typical of a pulsar wind nebula. The implied spin-down luminosity of the neutron star, assuming a conversion efficiency to nebular flux appropriate to Vela-like pulsars, is ~10^{35} ergs/s, again typical of objects a few tens of kyr old. Morphologically, the nebular flux is slightly enhanced along a direction, in projection on the sky, independently demonstrated to be of significance in radio polarization observations; we argue that this represents the orientation of the pulsar spin axis. At smaller scales, a narrow X-ray feature is seen extending out 5 from the point source, a distance consistent with the sizes of resolved wind termination shocks around many Vela-like pulsars. Finally, we argue based on synchrotron lifetimes in the estimated nebular magnetic field that DA 495 represents a rare pulsar wind nebula in which electromagnetic flux makes up a significant part, together with particle flux, of the neutron stars wind, and that this high magnetization factor may account for the nebulas low luminosity.