X-ray free-electron lasers (XFELs), with pulse durations of a few tens of femtoseconds or shorter, are cutting-edge instruments capable of observing structure and dynamics at the atomic scale. Temporal diagnostics are challenging but of fundamental importance for XFEL performance and experiments. In this paper, we demonstrate a method to characterize on a single-shot basis the temporal profile of both the electron beam and the XFEL radiation with femtosecond resolution. The approach consists in streaking the electron beam after the undulator using the wakefields of a corrugated structure. Its merits are arrival time stability and cost-effectiveness. Our method allows access to self-synchronized femtosecond diagnostics to any XFEL facility at low cost.