The current flowing across a semiconductor superlattice in tilted electric and magnetic fields is known to exhibit resonant enhancement, when Landau states of neighboring wells align at certain ratios of the field strengths. We show that the ultrafast version of this effect, in which coherent electron wave packets are involved, has a profound analogy to the Fiske effect in superconductor Josephson junctions and superfluid weak links, in that the coupling of the tunneling-induced charge oscillations (magneto-Bloch versus Josephson oscillations) to another oscillator (in-plane cyclotron oscillations versus external oscillator modes) opens an elastic rectifying transport channel. We explore the superlattice effect both theoretically and experimentally, and find that the transient self-induced current can be adequately modeled if the damping of both types of coupled electron oscillations is properly taken into account.