The central region of the Milky Way is one of the foremost locations to look for dark matter (DM) signatures. We report the first results on a search for DM particle annihilation signals using new observations from an unprecedented γ-ray survey of the Galactic Center (GC) region, i.e., the Inner Galaxy Survey, at very high energies (≳100 GeV) performed with the H.E.S.S. array of five ground-based Cherenkov telescopes. No significant γ-ray excess is found in the search region of the 2014-2020 dataset and a profile likelihood ratio analysis is carried out to set exclusion limits on the annihilation cross section ⟨σv⟩. Assuming Einasto and Navarro-Frenk-White (NFW) DM density profiles at the GC, these constraints are the strongest obtained so far in the TeV DM mass range. For the Einasto profile, the constraints reach ⟨σv⟩ values of 3.7×10^{-26} cm^{3} s^{-1} for 1.5 TeV DM mass in the W^{+}W^{-} annihilation channel, and 1.2×10^{-26} cm^{3} s^{-1} for 0.7 TeV DM mass in the τ^{+}τ^{-} annihilation channel. With the H.E.S.S. Inner Galaxy Survey, ground-based γ-ray observations thus probe ⟨σv⟩ values expected from thermal-relic annihilating TeV DM particles.