The recent detection of a three-hour X-ray flare from Sgr A* by Chandra provides very strong evidence for a compact emitting region near this supermassive black hole at the Galactic center. Sgr A*'s mm/sub-mm spectrum and linear polarimetric properties, and its quiescent-state X-ray flux density, are consistent with a model in which low angular momentum gas captured at large radii circularizes to form a hot, magnetized Keplerian flow within tens of Schwarzschild radii of the black hole's event horizon. In Sgr A*'s quiescent state, the X-ray emission appears to be produced by self-Comptonization (SSC) of the mm/sub-mm synchrotron photons emitted in this region. In this paper, we show that the prominent X-ray flare seen in Sgr A* may be due to a sudden enhancement of accretion through the circularized flow. Depending on whether the associated response of the anomalous viscosity is to increase or decrease in tandem with this additional injection of mass, the X-ray photons during the outburst may be produced either via thermal bremsstrahlung (if the viscosity decreases), or via SSC (if the viscosity increases). However, the latter predicts a softer X-ray spectrum than was seen by Chandra, so it appears that a bremsstrahlung origin for the X-ray outburst is favored. A strong correlation is expected between the mm/sub-mm and X-ray fluxes when the flare X-rays are produced by SSC, while the correlated variability is strongest between the sub-mm/far-IR and X-rays when bremsstrahlung emission is dominant during the flare. In addition, we show that future coordinated multi-wavelength observations planned for the 2002 and 2003 cycles may be able to distinguish between the accretion and jet scenarios.