Preliminary design of low-thrust trajectories in the circular restricted three-body problem (CR3BP) frequently relies upon ballistic dynamical structures and optimization algorithms. A fundamental understanding of how these dynamical structures change due to presence of a low-thrust force may lead to trajectories that cannot be obtained otherwise. This paper investigates the effect of a constant low-thrust acceleration on the horizontal Lyapunov (H-L) families in the CR3BP. Families of low-thrust periodic solutions are constructed in vicinity of L1 and L2 using numerical continuation methods. By either varying the Hamiltonian, acceleration magnitude, or acceleration orientation along the solution family, the effect of a low-thrust acceleration on H-L orbits is characterized. Investigating the geometry, bifurcations and hyperbolic unwinding behavior of these families provides insight into the low-thrust periodic solution structure of the Earth-Moon system. The introduction of a constant low-thrust acceleration distorts the geometry of ballistic H-L orbits into ’ear-shaped’ periodic solutions. The bifurcations of the low-thrust periodic solution families imply the existence of low-thrust halo, low-thrust axial, and low-thrust planar double-period families. Finally, low-thrust periodic solutions are identified that possess a higher rate of hyperbolic unwinding behavior than the ballistic L1 and L2 H-L families.