We present evidence for an X-ray jet in the T Tau star DG Tau A based on Chandra ACIS data. DG Tau A, a jet-driving classical T Tau star with a flat infrared spectrum, reveals an unusual X-ray spectrum that requires two thermal components with different intervening absorption column densities. The softer component shows a low temperature of T \approx 2.9 MK, and its absorption is compatible with the stellar optical extinction (hydrogen column density N_H \approx 5x10^21 cm^-2). In contrast, the harder component reveals a temperature (22 MK) characteristic for active T Tau stars but its emission is more strongly absorbed (N_H \approx 2.8x10^22 cm^-2). Furthermore, the high-resolution ACIS-S image reveals a weak excess of soft (0.5-2 keV) counts at distances of 2-4' from the star precisely along the optical jet, with a suggestive concentration at 4' where a bow-shock-like structure has previously been identified in optical line observations. The energy distribution of these photons is similar to those of the stellar soft component. We interpret the soft spectral component as originating from shocks at the base of the jet, with shock heating continuing out to a distance of at least 500 AU along the jet, whereas the hard component is most likely coronal/magnetospheric as in other young stellar systems.