We theoretically investigate resonant tunneling through a single level assisted by short LO phonon pulses. The analysis is based on the recently developed nonequilibrium linked-cluster expansion ͓P. Král, Phys. Rev. B 56, 7293 ͑1997͔͒, extended in this work to transient situations. The nonequilibrium spectral function for the resonance displays the formation and decay of the phonon sidebands on ultrashort time scales. The timedependent tunneling current through the individual phonon satellites reflects this quasiparticle formation by oscillations, whose time scale is set by the frequency of the phonon field and its harmonics. These oscillations are washed out at elevated temperatures. ͓S0163-1829͑99͒04208-3͔