Linear-response time-dependent density functional theory is used to investigate the relation between molecular bonding and surface plasmons for the model system H/Ag(111). We employ an orbital-dependent exchange-correlation functional to obtain a correct description of the Ag 3d band, which is crucial to avoid overscreening the plasmon by the s-d interband transitions. For the clean surface, this approach reproduces the experimental plasmon energies and dispersion to within 0.15 eV. Adsorption of hydrogen shifts and damps the Ag(111) surface plasmon and induces a new peak in the loss function at 0.6 eV below the Ag(111) plasmon peak. This feature originates from interband transitions between states located on the hydrogen atoms and states on the Ag surface atoms.