Fuel cells convert chemical energy directly into electrical energy with high efficiencies and environmental benefits, as compared with traditional heat engines. Yttria-stabilized zirconia is perhaps the material with the most potential as an electrolyte in solid oxide fuel cells (SOFCs), owing to its stability and near-unity ionic transference number. Although there exist materials with superior ionic conductivity, they are often limited by their ability to suppress electronic leakage when exposed to the reducing environment at the fuel interface. Such electronic leakage reduces fuel cell power output and the associated chemo-mechanical stresses can also lead to catastrophic fracture of electrolyte membranes. Here we depart from traditional electrolyte design that relies on cation substitution to sustain ionic conduction. Instead, we use a perovskite nickelate as an electrolyte with high initial ionic and electronic conductivity. Since many such oxides are also correlated electron systems, we can suppress the electronic conduction through a filling-controlled Mott transition induced by spontaneous hydrogen incorporation. Using such a nickelate as the electrolyte in free-standing membrane geometry, we demonstrate a low-temperature micro-fabricated SOFC with high performance. The ionic conductivity of the nickelate perovskite is comparable to the best-performing solid electrolytes in the same temperature range, with a very low activation energy. The results present a design strategy for high-performance materials exhibiting emergent properties arising from strong electron correlations.