Recently, the electric drift current bias was introduced as a new paradigm to break the Lorentz reciprocity in graphene. Here, we study the impact of the nonreciprocal response of such a solution in the context of the Cherenkov problem and investigate the emission of light by a pencil beam of moving charges in a graphene sheet with drifting electrons. It is demonstrated that the drift bias enables strongly asymmetry Cherenkov emissions with spectrum dependent on the sign of the charges velocity. The key role of the unidirectional graphene plasmons in this process is highlighted.