Ultrafast, high-efficiency single-photon detectors are among the most sought-after elements in modern quantum optics and quantum communication. However, imperfect modal matching and finite photon absorption rates have usually limited their maximum attainable detection efficiency. Here we demonstrate superconducting nanowire detectors atop nanophotonic waveguides, which enable a drastic increase of the absorption length for incoming photons. This allows us to achieve high on-chip single-photon detection efficiency up to 91% at telecom wavelengths, repeatable across several fabricated chips. We also observe remarkably low dark count rates without significant compromise of the on-chip detection efficiency. The detectors are fully embedded in scalable silicon photonic circuits and provide ultrashort timing jitter of 18 ps. Exploiting this high temporal resolution, we demonstrate ballistic photon transport in silicon ring resonators. Our direct implementation of a high-performance single-photon detector on chip overcomes a major barrier in integrated quantum photonics.