We report on high-efficiency superconducting nanowire single-photon detectors based on amorphous tungsten silicide and optimized at 1064 nm. At an operating temperature of 1.8 K, we demonstrated a 93% system detection efficiency at this wavelength with a dark noise of a few counts per second. Combined with cavity-enhanced spontaneous parametric downconversion, this fiber-coupled detector enabled us to generate narrowband single photons with a heralding efficiency greater than 90% and a high spectral brightness of 0.6×104 photons/(s·mW·MHz). Beyond single-photon generation at large rate, such high-efficiency detectors open the path to efficient multiple-photon heralding and complex quantum state engineering.