We study experimentally the possibility to utilize pulses of pure spin current, produced via the nonlocal spin injection mechanism, to generate short packets of spin waves propagating in nanoscale magnetic waveguides. Spatially and time-resolved micro-focus Brillouin light scattering spectroscopy measurements demonstrate that the excitation by spin current results in extremely fast transient response, enabling efficient generation of short spin-wave packets with duration down to a few nanoseconds. The proposed method opens a route for the implementation of high-speed magnonic systems for transmission and processing of information on the nanoscale. Published by AIP Publishing.