Using the ferroin-catalyzed Belosov-Zhabotinsky (BZ) reaction with the addition of an activating reactant in a reaction-diffusion system, we experimentally studied the competition between target waves and spiral waves when the system was in an oscillatory state. We found that when the period of background oscillation is smaller than that of spiral waves, target waves automatically appear in the reaction medium despite of the presence of spiral waves. The target waves compete with spiral waves and eventually win over the spirals, driving them out of the reaction medium. We also carried out numerical simulations using a modified Oregonator model. Our simulation results qualitatively agree with the experimental observations.