We investigate the nature of the time-reversal breaking pairing state in the hole-doped monolayer MoS2 on the basis of the realistic three-orbital attractive Hubbard-like model with the atomic spin–orbit coupling. Due to the multi-orbital features arising from the Mo d orbitals in the noncentrosymmetric crystal structure, the Lifshitz transition takes place upon hole doping. Across the Lifshitz transition point, the sign of the relative phase between the Cooper-pair components drastically changes, leading to the emergence of the time-reversal breaking phase with complex gap functions. It is shown that this intriguing pairing state is characterized by the finite momentum-space distributions of the orbital and spin angular momentum with three-fold rotational symmetry on the Fermi-surface pockets around K and KA points. The present analysis based on the realistic multi-orbital model demonstrates that the spin–orbit-coupled metals in noncentrosymmetric crystal structures have favorable conditions for the realization of the time-reversal breaking superconductivity.