Future exploration of the Moon will involve numerous both crewed and uncrewed launches, essential for transporting materials and instruments critical to the success of lunar missions. The L₁ Lagrangian point can represent an ideal transit hub for space missions due to its dynamical properties and strategic location between Earth and Moon. This paper shows an efficient and systematic technique based on the theory of functional connections to evaluate transfers from Earth to Moon, making use of the stable/unstable manifolds associated with the family of Lyapunov orbits around L₁. Tens of millions of transfers are evaluated, narrowing the space of solutions to the ones with the lowest costs in an extensive investigation. The transfer is designed under the circular restricted three-body problem formulation with a specific system of coordinates and constraints. A Moon flyby is followed by an entry into a Lyapunov orbit around L₁, from where a final transfer to the Moon is performed, with every leg aided by an invariant manifold. The results show a reduced fuel consumption of at least 58.80 m/s as compared to the literature. This difference can be useful for mission designers, while the efficiency of the technique can potentially extend the results to other types of transfers in space.