Combining photoelectron spectroscopy with tunable laser pulse excitation allows us to characterize the Coulomb barrier potential of multiply negatively charged silver clusters. The spectra of mass- and charge-selected polyanionic systems, with z=2-5 excess electrons, show a characteristic dependence on the excitation energy, which emphasizes the role of electron tunneling through the barrier. By evaluating experimental data from an 800-atom system, the electron yield is parametrized with respect to tunneling near the photoemission threshold. This analysis results in the first experimentally based potential energy functions of polyanionic metal clusters.