Particle accelerators represent an indispensable tool in science and industry. However, the size and cost of conventional radio-frequency accelerators limit the utility and reach of this technology. Dielectric laser accelerators (DLAs) provide a compact and cost-effective solution to this problem by driving accelerator nanostructures with visible or near-infrared pulsed lasers, resulting in a 104 reduction of scale. Current implementations of DLAs rely on free-space lasers directly incident on the accelerating structures, limiting the scalability and integrability of this technology. We present an experimental demonstration of a waveguide-integrated DLA that was designed using a photonic inverse-design approach. By comparing the measured electron energy spectra with particle-tracking simulations, we infer a maximum energy gain of 0.915 kilo-electron volts over 30 micrometers, corresponding to an acceleration gradient of 30.5 mega-electron volts per meter. On-chip acceleration provides the possibility for a completely integrated mega-electron volt-scale DLA.