We report on the observation of edge electric currents excited in bilayer graphene by terahertz laser radiation. We show that the current generation belongs to the class of second order in electric field phenomena and is controlled by the orientation of the THz electric field polarization plane. Additionally, applying a small magnetic field normal to the graphene plane leads to a phase shift in the polarization dependence. With increasing the magnetic field strength, the current starts to exhibit 1/B-magneto-oscillations with a period consistent with that of the Shubnikov-de Haas effect and amplitude by an order of magnitude larger as compared to the current at zero magnetic field measured under the same conditions. The microscopic theory developed shows that the current is formed in the edge's vicinity limited by the mean-free path of carriers and the screening length of the high-frequency electric field. The current originates from the alignment of the free carrier momenta and dynamic accumulation of charge at the edges, where the P-symmetry is naturally broken. The observed magneto-oscillations of the photocurrent are attributed to the formation of Landau levels.