We show by the first-order perturbation theory and the configuration interaction method that the Coulomb interaction in quantum rings mixes electron-hole pair states with the same total angular momentum, which makes it difficult to observe a clear excitonic Aharonov-Bohm (A-B) effect. To avoid this situation, we propose the use of a combined structure of a quantum dot on the top of a quantum ring with an applied static electric field. Under moderate experimental conditions with respect to the applied electric and magnetic fields, we show that we can observe the excitonic A-B effect due to the reduction of the Coulomb interaction and an increase in the difference between the average radii of the electron and hole trajectories.