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Temporal modes (TMs) of photons provide an appealing high-dimensional encoding basis for quantum information. While techniques to generate TM states have been established, high-dimensional decoding of single-photon TMs remains an open challenge. In this work, we experimentally demonstrate demultiplexing of five-dimensional TMs of single photons with an average fidelity of 0.96±0.01, characterized via measurement tomography. This is achieved with use of a newly developed device, the multi-output quantum pulse gate (MQPG). We demonstrate a proof-of-principle complete decoder based on the MQPG that operates on any basis from a set of six five-dimensional mutually unbiased bases and is therefore suitable as a receiver for high-dimensional quantum key distribution. Furthermore, we confirm the high-quality operation of the MQPG by performing resource-efficient state tomography with an average fidelity of 0.98±0.02.