We study theoretically the light-induced magnetization switching in a binary ferrimagnet of the type ${\mathrm{A}}_{p}{\mathrm{B}}_{1\ensuremath{-}p}$, randomly occupied by two different species of magnetic ions. The localized spins are coupled with spins of itinerant electrons via the $s\text{\ensuremath{-}}d$ exchange interaction. The dynamics of the localized and itinerant spins is described by coupled rate equations, which include electron-phonon interaction, spin-lattice relaxation, and exchange scattering, induced by the $s\text{\ensuremath{-}}d$ exchange interaction. The exchange scattering leads to the formation of ferromagneticlike states at initial temperatures $T$ both below and above the magnetization compensation temperature ${T}_{M}$ with the opposite polarities in these two temperature regions. Inclusion of electron-phonon interaction and spin-lattice relaxation in the dynamical equations leads to the switching in a temperature range $0lTl{T}_{f}$, where ${T}_{f}$ is slightly higher than ${T}_{M}$ and strongly depends on the spin-lattice relaxation time of itinerant electrons. The switching requires less pulse fluence in the vicinity of ${T}_{M}$.