Recent analysis of data sets from two extensive air shower cosmic-ray detectors shows tantalizing evidence of an anisotropic overabundance of cosmic rays toward the Galactic center region that "turns on" around 10¹⁸ eV. We demonstrate that the anisotropy could be due to neutrons created in the Galactic center region through charge exchange in proton-proton collisions, where the incident, high-energy protons obey a ~E^-2 power law associated with acceleration at a strong shock. We show that the normalization supplied by the gamma-ray signal from EGRET GC source 3EG J1746-2851 (ascribed to p-p-induced neutral pion decay at GeV energies), together with a very reasonable spectral index of 2.2, predicts a neutron flux at ~10¹⁸ eV fully consistent with the extremely high energy cosmic-ray data. Likewise, the normalization supplied by the very recent GC data from the HESS air Cerenkov telescope at ~TeV energies is almost equally compatible with the ~10¹⁸ eV cosmic-ray data. Interestingly, however, the EGRET and HESS data appear to be themselves incompatible. We find a plausible resolution of this discrepancy in an effective two-source model. Finally, we argue that the shock acceleration is probably occurring in the shell of Sagittarius A East, an unusual supernova remnant located very close to the Galactic center. In support of this contention we note that (1) the extended shell of this object could provide both of the sources suggested by the gamma-ray data and (2) the unusually strong magnetic field at this remnant, together with a perpendicular shock geometry, allow for acceleration of protons up to the extreme energies required to explain the cosmic-ray anisotropy. If the connection between the anisotropy and Sagittarius A East could be firmly established, it would be the first direct evidence for a particular Galactic source of cosmic rays up to energies near 10¹⁹ eV.