We predict streaming current inversion with multivalent counterions in hydrodynamically driven polymer translocation events from a correlation-corrected charge transport theory including charge fluctuations around mean-field electrostatics. In the presence of multivalent counterions, electrostatic many-body effects result in the reversal of the DNA charge. The attraction of anions to the charge-inverted DNA molecule reverses the sign of the ionic current through the pore. Our theory allows for a comprehensive understanding of the complex features of the resulting streaming currents. The underlying mechanism is an efficient way to detect DNA charge reversal in pressure-driven translocation experiments with multivalent cations.