In recent years, near-IR and X-ray flares have been detected from the Galaxy's central radio point source, Sagittarius A* (Sgr A*), believed to be a \~3.10^6 solar masses supermassive black hole. In some cases, the transient emission appears to be modulated with a (quasi-)periodic oscillation (QPO) of ~ 17-20 minutes. The implied ~ 3 r_S size of the emitter (where r_S = 2GM/c^2 is the Schwarzschild radius) points to an instability - possibly induced by accretion - near the Marginally Stable Orbit (MSO) of a slowly spinning object. But Sgr A* is not accreting via a large, 'standard' disk; instead, the low density environment surrounding it apparently feeds the black hole with low angular momentum clumps of plasma that circularize within ~ 10-300 r_S and merge onto a compact, hot disk. In this Letter, we follow the evolution of the disk following such an event, and show that a Rossby wave instability, particularly in its magnetohydrodynamic (MHD) form, grows rapidly and produces a period of enhanced accretion lasting several hours. Both the amplitude of this response, and its duration, match the observed flare characteristics rather well.