Using first-principles calculations based on density-functional theory (DFT) we investigated the effects of the van der Waals (vdW) interactions on the structural and electronic properties of anthracene and pentacene adsorbed on the Ag(111) surface. We found that the inclusion of vdW corrections strongly affects the binding of both anthracene/Ag(111) and pentacene/Ag(111), yielding adsorption heights and energies more consistent with the experimental results than standard DFT calculations with generalized gradient approximation (GGA). For anthracene/Ag(111) the effect of the vdW interactions is even more dramatic: we found that "pure" DFT-GGA calculations (without including vdW corrections) result in preference for a tilted configuration, in contrast to experimental observations of flat-lying adsorption; including vdW corrections, on the other hand, alters the binding geometry of anthracene/Ag(111), favoring the flat configuration. The electronic structure obtained using a self-consistent vdW scheme was found to be nearly indistinguishable from the conventional DFT electronic structure once the correct vdW geometry is employed for these physisorbed systems. Moreover, we show that a vdW correction scheme based on a hybrid functional DFT calculation (HSE) results in an improved description of the highest occupied molecular level of the adsorbed molecules.