Trap states in semiconductors usually degrade charge separation and collection in photovoltaics due to trap-mediated nonradiative recombination. Here, it is found that perovskite can be heavily doped in low concentration with non-ignorable broadband infrared absorption in thick films and their trap states accumulate electrons through infrared excitation and hot carrier cooling. A hybrid one-sided abrupt perovskite/TiO2 p-N heterojunction is demonstrated that enables partial collection of these trap-filled charges through a tunneling process instead of detrimental recombination. The tunneling is from broadband trap states in the wide depleted p-type perovskite, across the barrier of the narrow depleted TiO2 region (<5 nm), to the N-type TiO2 electrode. The trap states inject carriers into TiO2 through tunneling and produce aroundunity peak external quantum efficiency, giving rise to near-infrared photovoltaics. The near-infrared response allows photodetecting devices to work in both diode and conductor modes. This work opens a new avenue to explore the near-infrared application of hybrid perovskites.