Based on model calculations, we predict a magnetoelectric tunneling electroresistance effect in multiferroic tunnel junctions consisting of ferromagnetic electrodes and magnetoelectric antiferromagnetic barriers. Switching of the antiferromagnetic order parameter in the barrier in applied electric field by means of the magnetoelectric coupling leads to a substantial change of the resistance of the junction. The effect is explained in terms of the switching of the orientations of local magnetizations at the barrier interfaces affecting the spin-dependent interface transmission probabilities. Magnetoelectric multiferroic materials with finite ferroelectric polarization exhibit an enhanced resistive change due to polarization-induced spin-dependent screening. These results suggest that devices with active barriers based on single-phase magnetoelectric antiferromagnets represent an alternative nonvolatile memory concept.