The non-thermal nature of the X-ray emission from the shell-type supernova remnants (SNRs) G1.9+0.3 and G330.2+1.0 is an indication of intense particle acceleration in the shock fronts of both objects. This suggests that the SNRs are prime candidates for very-high-energy (VHE; E > 0.1 TeV) γ-ray observations. G1.9+0.3, recently established as the youngest known SNR in the Galaxy, also offers a unique opportunity to study the earliest stages of SNR evolution in the VHE domain. The purpose of this work is to probe the level of VHE γ-ray emission from both SNRs and use this to constrain their physical properties. Observations were conducted with the H.E.S.S. (High Energy Stereoscopic System) Cherenkov Telescope Array over a more than six-year period spanning 2004–2010. The obtained data have effective livetimes of 67 h for G1.9+0.3 and 16 h for G330.2+1.0. The data are analysed in the context of the multiwavelength observations currently available and in the framework of both leptonic and hadronic particle acceleration scenarios. No significant γ-ray signal from G1.9+0.3 or G330.2+1.0 was detected. Upper limits (99 per cent confidence level) to the TeV flux from G1.9+0.3 and G330.2+1.0 for the assumed spectral index Γ = 2.5 were set at 5.6 × 10−13 cm−2 s−1 above 0.26 TeV and 3.2 × 10−12 cm−2 s−1 above 0.38 TeV, respectively. In a one-zone leptonic scenario, these upper limits imply lower limits on the interior magnetic field to B G1.9 ≳ 12 μG for G1.9+0.3 and to B G330 ≳ 8 μG for G330.2+1.0. In a hadronic scenario, the low ambient densities and the large distances to the SNRs result in very low predicted fluxes, for which the H.E.S.S. upper limits are not constraining.