The recent detection of TeV gamma-radiation from the direction of the Galactic Center within several arc-minutes around Sgr A* is the first model-independent evidence of existence of high-energy particle accelerator(s) in the central 10 pc region of our Galaxy. This is an extraordinary site that harbours many remarkable objects with the compact radio source Sgr A* – a hypothetical super-massive black hole (SMBH) – in the dynamical center of the Galaxy. Here we explore the possible direct and indirect links of the reported TeV emission to the SMBH. We show that at least three γ-ray production scenarios that take place close to the event horizon of the SMBH can explain the reported TeV fluxes. An alternative (or additional) channel of TeV radiation is related to the run-away protons accelerated in Sgr A*. Quasi-continuous injection of relativistic protons into the surrounding dense gas environment initiates detectable high-energy gamma-ray emission. The absolute flux and the energy spectrum of this radiation component strongly depend on the history of particle injection and the character of diffusion of protons during the last 105 yr. For a reasonable combination of a few model parameters, one can explain the detected gamma-ray flux solely by this diffuse component.