Nuclear masses play a fundamental role in understanding how the heaviest elements in the Universe are created in the r process. We predict r-process nucleosynthesis yields using neutron capture and photodissociation rates that are based on the nuclear density functional theory. Using six Skyrme energy density functionals based on different optimization protocols, we determine for the first time systematic uncertainty bands-related to mass modeling-for r-process abundances in realistic astrophysical scenarios. We find that features of the underlying microphysics make an imprint on abundances especially in the vicinity of neutron shell closures: Abundance peaks and troughs are reflected in trends of neutron separation energy. Further advances in the nuclear theory and experiments, when linked to observations, will help in the understanding of astrophysical conditions in extreme r-process sites.