Background: Inelastic neutrino-nucleus scattering through the weak neutral-current plays an important role in a stellar environment where the transport of neutrinos determines the rate of cooling. Since there are no direct experimental data on neutral-current neutrino-nucleus cross sections available, only the modeling of these reactions provides the relevant input for supernova simulations. Purpose: To establish a fully self-consistent framework for neutral-current neutrino-nucleus reactions based on a relativistic nuclear energy density functional. Methods: Neutrino-nucleus cross sections are calculated using a weak Hamiltonian and nuclear properties of initial and excited states are obtained with a relativistic Hartree-Bogoliubov model and a relativistic quasiparticle random phase approximation that is extended to include pion contributions for unnatural parity transitions. Results: Inelastic neutral-current neutrino-nucleus cross sections for 12C, 16O, 56Fe, 56Ni, and even isotopes 92−100Mo as well as respective cross sections averaged over distribution of supernova neutrinos. Conclusions: The present study provides insight into neutrino- nucleus scattering cross sections in the neutral channel, their theoretical uncertainty in view of recently developed microscopic models, and paves the way for systematic self-consistent large- scale calculations involving open-shell target nuclei.