We have investigated the nature of flare emission from Sgr A* during multi-wavelength observations of this source that took place in 2004, 2005, and 2006. We present evidence for dimming of submillimeter and radio flux during the peak of near-IR flares. This suggests that the variability of Sgr A* across its wavelength spectrum is phenomenologically related. The model explaining this new behavior of flare activity could be consistent with adiabatically cooling plasma blobs that are expanding but also partially eclipsing the background quiescent emission from Sgr A*. When a flare is launched, the plasma blob is most compact and is brightest in the optically thin regime whereas the emission in radio/submillimeter wavelengths has a higher opacity. Absorption in the observed light curve of Sgr A* at radio/submillimeter flux is due to the combined effects of lower brightness temperature of plasma blobs with respect to the quiescent brightness temperature and high opacity of plasma blobs. This implies that plasma blobs are mainly placed in the magnetosphere of a disk-like flow or further out in the flow. The depth of the absorption being larger in submillimeter than in radio wavelengths implies that the intrinsic size of the quiescent emission increases with increasing wavelength which is consistent with previous size measurements of Sgr A*. Lastly, we believe that occultation of the quiescent emission of Sgr A* at radio/submillimeter by IR flares can be used as a powerful tool to identify flare activity at its earliest phase of its evolution.