Despite strong physical reasons that they should exist and decades of searching, jets from the Galactic center black hole, Sgr A*, have not yet been convincingly detected. Based on high-resolution Very Large Array images and ultra-deep imaging-spectroscopic data produced by the Chandra X-ray Observatory, we report new evidence for the existence of a parsec-scale jet from Sgr A*, by associating a linear feature G359.944-0.052, previously identified in X-ray images of the Galactic center, with a radio shock front on the Eastern Arm of the Sgr A West H II region. We show that the shock front can be explained in terms of the impact of a jet having a sharp momentum peak along the Galaxy's rotation axis, whereas G359.944-0.052, a quasi-steady feature with a power-law spectrum, can be understood as synchrotron radiation from shock-induced ultra-relativistic electrons cooling in a finite post-shock region downstream along the jet path. Several interesting implications of the jet properties are discussed. © 2013. The American Astronomical Society. All rights reserved..