We study the response of exciton complexes in semiconductor quantum rings to vertical electric fields, using path integral quantum Monte Carlo simulations. The interaction of a vertical applied field and the piezoelectric fields of the ring with strongly-correlated excitonic complexes switches excitons between two different lateral locations within the ring . This control should be observable through polarizability and dipole measurements, and, for biexcitons, an energy shift beyond the normal Stark shift.