Several nearby ultra-faint satellites of the Milky Way discovered by the Dark Energy Survey (DES) during the last few years are promising targets for indirect dark matter (DM) searches with very-high-energy (VHE, E$\gtrsim$100 GeV) gamma rays. The H.E.S.S. experiment has carried out an observational campaign in 2017 and 2018 towards a selection of the most promising DES dwarf satellites, accumulating a total observation time of about 80 hours. The individual datasets have been used to look for a DM signal in several annihilation channels. No significant VHE gamma-ray excess above the background is found in any of the targets. Constraints are derived on the thermally-averaged velocity-weighted annihilation cross section $\langle \sigma v \rangle$ of the DM particles versus their mass. A combined analysis of the datasets has been performed. The strongest 95\% C.L. upper limits reach a $\langle \sigma v \rangle$ value of a few $10^{-24}$ cm$^3$ s$^{-1}$ for the DM continuum signals in the TeV DM mass range. They are the first constraints derived from imaging atmospheric Cherenkov telescope observations towards DES dwarf galaxy satellites. These limits are among the most constraining so far in the TeV DM mass range towards dwarf satellites of the Milky Way.