The inner region of the Milky Way halo harbors a large amount of dark matter (DM). Given its proximity, it is one of the most promising targets to look for DM. We report on a search for the annihilations of DM particles using γ-ray observations towards the inner 300 pc of the Milky Way, with the H.E.S.S. array of ground-based Cherenkov telescopes. The analysis is based on a 2D maximum likelihood method using Galactic Center (GC) data accumulated by H.E.S.S. over the last 10 years (2004-2014), and does not show any significant γ-ray signal above background. Assuming Einasto and Navarro-Frenk-White DM density profiles at the GC, we derive upper limits on the annihilation cross section ⟨σv⟩. These constraints are the strongest obtained so far in the TeV DM mass range and improve upon previous limits by a factor 5. For the Einasto profile, the constraints reach ⟨σv⟩ values of 6×10^{-26} cm^{3} s^{-1} in the W^{+}W^{-} channel for a DM particle mass of 1.5 TeV, and 2×10^{-26} cm^{3} s^{-1} in the τ^{+}τ^{-} channel for a 1 TeV mass. For the first time, ground-based γ-ray observations have reached sufficient sensitivity to probe ⟨σv⟩ values expected from the thermal relic density for TeV DM particles.