A TeV scale electroweak particle is a well motivated candidate for the dark matter (DM) of our Universe. Yet such a particle may only be detectable using indirect detection instruments sensitive to TeV-scale gamma rays that can result from dark matter annihilations. We present a mock analysis of the sensitivity for the present ground-based Cherenkov telescope array H.E.S.S. (High Energy Spectroscopic System) to detect TeV scale DM in the Galactic Center region. The work combines next-to-leading-logarithmic order calculations for the annihilation photon spectrum, as well as a comprehensive treatment of detector effects and expected backgrounds. Forecast limits on the sensitivity of H.E.S.S. have been derived across the important TeV mass range, assuming different DM density profiles and focusing on the canonical WIMP dark matter candidate Wino. These limits test our present and future ability to probe the predicted thermal cross section for some of the most promising DM candidates that could be discovered in the coming decade.