We treat quench and out-of-equilibrium dynamics of a single-molecule device, modeled by an individual resonant level weakly coupled by displacement to a molecular vibronic mode. Mapping the model onto a continuum-limit Hamiltonian, we use Abelian bosonization to write the latter in a form quadratic in bosonic operators. The resulting Hamiltonian is exactly solvable, allowing a precise calculation of the quench dynamics in response to a sudden change in the system parameters. Several quench scenarios are considered, including an abrupt switching on of the electron-phonon interaction and a sudden change in the phonon frequency. We are also able to calculate the exact response to an ac drive. The resulting dynamics in each of these cases will be discussed, with special emphasize on the underlying time scales involved.