We show that magnetic fields significantly enhance a new tunneling mechanism in quantum field theories with photons coupling to fermionic minicharged particles (MCPs). We propose a dedicated laboratory experiment of the light-shining-through-walls type that can explore a parameter regime comparable to and even beyond the best model-independent cosmological bounds. With present-day technology, such an experiment is particularly sensitive to MCPs with masses in and below the meV regime as suggested by new-physics extensions of the standard model.