We introduce a simple and flexible concept for a heralded—spectrally pure—single-photon source. The scheme uses a probabilistic photon-pair source pumped with a continuous-wave laser, whereby a rapid gating InGaAs/InP single-photon avalanche diode provides a synchronous clock for the system. The low timing jitter of the detector temporally resolves, and hence spectrally filters, the heralded photons. We demonstrate the concept by combining this with a narrow-band integrated silicon nitride photon-pair source. This simple architecture is capable of heralding photons with high spectral purity in the telecom band but could be adapted to other wavelengths and bandwidth regimes.