The three-body scattering problem in Coulombic systems is a widespread, yet unresolved problem using the mathematically rigorous methods. In this work this long-term challenge has been undertaken by combining distorted waves and Faddeev–Merkuriev equation formalisms in conjunction with the complex scaling technique to overcome the difficulties related with the boundary conditions. Unlike the common belief, it is demonstrated that the smooth complex scaling method can be applied to solve the three-body Coulomb scattering problem in a wide energy region, including the fully elastic domain and extending to the energies well beyond the atom ionization threshold. A newly developed method is used to study electron scattering on the ground states of hydrogen and positronium atoms as well as a reaction. Where available, obtained results are compared with the experimental data and theoretical predictions, proving the accuracy and efficiency of the newly developed method.