Thermal dark matter at the MeV mass scale has its abundance set during the highly nontrivial epochs of neutrino decoupling and electron annihilation. The technical obstacles attached to solving Boltzmann equations of multiple interacting sectors being both relativistic and nonrelativistic have to-date prevented the full treatment of this problem. Here, for the first time, we calculate the freeze-out of light dark matter, taking into account the energy transfer between the dark sector, neutrinos, and the electromagnetically interacting plasma from annihilation and elastic scattering processes alike. We develop a numerically feasible treatment that allows to track photon and neutrino temperatures across freeze-out and to arrive at a precision prediction of $N_{\mathrm{eff}}$ for arbitrary branching ratios of the dark matter annihilation channels. In addition, our treatment resolves for the first time the dark matter temperature evolution across freeze-out involving three sectors. It enters in the efficiency of velocity-dependent annihilation channels and for a flavor-blind $p$-wave annihilation into electron- and neutrino-pairs of all generations, we find the present Planck data exclude a complex scalar dark matter particle of mass of $m_{\varphi }\le 7\mathrm{MeV}$.