We theoretically analyze the elctron dynamics in a superlattice (SL) driven by a highly asymmetric electromagnetic pulse. We show that the time asymmetry of the pulse results in indirect population of higher minibands and leads to the appearance of a direct current. At low temperatures and low filling for the lowest miniband the current generated in the SL has an oscillating behavior as a function of the pulse strength with minima occurring when the wave vector transferred by the pulse matches an integer of half the reciprocal lattice constant.