Young core-collapse supernovae with dense-wind progenitors may be able to accelerate cosmic-ray hadrons beyond the knee of the cosmic-ray spectrum, and this may result in measurable gamma-ray emission. We searched for gamma-ray emission from ten supernovae observed with the High Energy Stereoscopic System (H.E.S.S.) within a year of the supernova event. Nine supernovae were observed serendipitously in the H.E.S.S. data collected between December 2003 and December 2014, with exposure times ranging from 1.4 to 53 h. In addition we observed SN 2016adj as a target of opportunity in February 2016 for 13 h. No significant gamma-ray emission has been detected for any of the objects, and upper limits on the >1 TeV gamma-ray flux of the order of ~10⁻¹³ cm⁻²s⁻¹ are established, corresponding to upper limits on the luminosities in the range ~2 × 10³⁹ to ~1 × 10⁴² erg s⁻¹. These values are used to place model-dependent constraints on the mass-loss rates of the progenitor stars, implying upper limits between ~2 × 10⁻⁵ and ~2 × 10⁻³ M_⊙ yr⁻¹ under reasonable assumptions on the particle acceleration parameters.