A system-specific but very accurate density functional theory (DFT) correction scheme is proposed for precise calculations of adsorbent-adsorbate interactions by combining the non-empirical van der Waals density functional (vdW-DF) method and the empirical DFT/CC correction scheme to reach accuracy of the coupled clusters method with single, double and perturbative triple excitations (CCSD(T)). The new approach is applied to small molecules (CH4, CO2, H2, H2O, N2) interacting with silica surfaces and purely siliceous microporous solids. The vdW-DF/CC results for a perfectly reconstructed α-quartz surface are consistent with other dispersion-corrected DFT methods. Corrected for ZPVE, the vdW-DF/CC enthalpies of adsorption in pure-silica zeolite LTA (ΔHads(0 K)) of 3.6 and 5.2 kcal/mol for methane and carbon dioxide, respectively, are in excellent agreement with experimental values of 3.6 and 5.0 kcal/mol. The very high accuracy of the new scheme and its relatively easy use and numerical stability as compared to the earlier DFT/CC scheme offer a straightforward solution for obtaining reliable predictions of adsorption energies.