'More than Moore' captures a concept for overcoming limitations in silicon electronics by incorporating new functionalities in the constituent materials. Perovskite oxides are candidates because of their vast array of physical properties in a common structure. They also enable new electronic devices based on strongly-correlated electrons. The field effect transistor and its derivatives have been the principal oxide devices investigated thus far, but another option is available in a different geometry: if the current is perpendicular to the interface, the strong internal electric fields generated at back-to-back heterojunctions can be used for oxide electronics, analogous to bipolar transistors. Here we demonstrate a perovskite heteroepitaxial metal-base transistor operating at room temperature, enabled by interface dipole engineering. Analysis of many devices quantifies the evolution from hot-electron to permeable-base behaviour. This device provides a platform for incorporating the exotic ground states of perovskite oxides, as well as novel electronic phases at their interfaces.