Near-Earth Asteroids (NEA) tell us a lot of the early Solar System. Space missions aimed to those objects can provide information for a better understanding of physical and chemical processes of Earth's formation. In this perspective, it is necessary to invest in optimized techniques orbital transfer able to achieve these objects with reduced spent fuel. In this paper, we explore the dynamics of unstable periodic orbits around the Lagrangian point L1 and the gravitational influence of the moon in order to get the energy needed to overcome the gravity of the Earth-Moon system and reach a NEA. The fuel consumption in this kind of maneuver is less than that required in other approaches. The escape trajectories obtained present sensitive dependence on initial conditions and can be judiciously controlled by small perturbations so that they are targeted to specific regions of the Solar System, being propitious mainly for missions whose targets belong to Apollos and Atens classes.