In recent decades, Dengue fever and its deadly complications, such as Dengue hemorrhagic fever, have become one of the major mosquito-transmitted diseases, with an estimate of 390 million cases occurring annually in over 100 tropical and subtropical countries, most of which belonging to the developing world. Empirical evidence indicates that the most effective mechanism to reduce Dengue infections is to combat the disease-carrying vector, which is often implemented via chemical pesticides to destroy mosquitoes in their adult or larval stages. The present paper considers an SIR epidemiological model describing the vector-to-host and host-to-vector transmission dynamics. The model includes pesticide control represented in terms of periodic impulsive perturbations, as well as seasonal fluctuations of the vector growth and transmission rates of the disease. The effectiveness of the control strategy is studied numerically in detail by means of path-following techniques for non-smooth dynamical systems. Special attention is given to determining the optimal timing of the pesticide applications, in such a way that the number of infections and the required amount of pesticide are minimized.