Semiconductor quantum dots in photonic integrated circuits enable scaling quantum-information processing to many single photons and quantum-optical gates. Obtaining high-purity and coherent single photons from quantum dots requires spectral filtering to select individual excitonic transitions. Here, an on-chip wavelength-tunable filter integrated with a single-photon source, which preserves the optical properties of the emitter, is demonstrated. Nanomechanical motion is used for tuning the resonant wavelength over 10 nm, enabling operation at cryogenic temperatures, and single-photon emission from a quantum dot under non-resonant excitation is demonstrated without resorting to free-space optical filters. These results are key for the development of fully integrated de-multiplexing, multi-path photon encoding schemes, and multi-emitter circuits.