Net acceleration of attosecond-scale electron pulses is critical to the development of on-chip accelerators. We demonstrate a silicon-based laser-driven two-stage accelerator as an injector stage prototype for a Dielectric Laser Accelerator (DLA). The first stage converts a 57-keV (500±100)-fs (FWHM) electron pulse into a pulse train of 700±200 as (FWHM) microbunches. The second stage harnesses the tunability of dual-drive DLA to perform both a net acceleration and a streaking measurement. In the acceleration mode, the second stage increases the net energy of the electron pulse by 200 eV over 12.25 μm. In the deflection mode, the microbunch temporal profile is analyzed by a direct streaking measurement with 200 as resolution. This work provides a demonstration of a novel, on-chip method to access the attosecond regime, opening new paths towards attosecond science using DLA.