Superconducting nanowire single-photon detectors (SNSPDs) provide high efficiency for detecting individual photons while keeping dark counts and timing jitter minimal. Besides superior detection performance over a broad optical bandwidth, compatibility with an integrated optical platform is a crucial requirement for applications in emerging quantum photonic technologies. Here we present SNSPDs embedded in nanophotonic integrated circuits which achieve internal quantum efficiencies close to unity at 1550 nm wavelength. This allows for the SNSPDs to be operated at bias currents far below the critical current where unwanted dark count events reach milli-Hz levels while on-chip detection efficiencies above 70% are maintained. The measured dark count rates correspond to noise-equivalent powers in the 10-19 W/Hz-1/2 range and the timing jitter is as low as 35 ps. Our detectors are fully scalable and interface directly with waveguide-based optical platforms.