We report on a comprehensive experimental and theoretical study of optical third harmonic generation (THG) on the exciton-polariton resonances in the zinc-blende semiconductors GaAs, CdTe, and ZnSe subject to an external magnetic field, representing a topic that had remained unexplored so far. In these crystals, crystallographic THG is allowed in the electric-dipole approximation, so that substantial magnetic-field-induced changes of the THG are unexpected: the symmetry reduction due to magnetic field, corresponding change of the selection rules, and the Zeeman effect are expected to play a minor role. Surprisingly, we observe a strong enhancement of the THG intensity by a factor of 50 for the 1s exciton-polariton in GaAs in magnetic fields up to 10 T. In contrast, the corresponding enhancement is moderate in CdTe and almost absent in ZnSe. In order to explain this strong variation, we develop a microscopic theory accounting for the optical harmonics generation on exciton-polaritons and analyze the THG mechanisms induced by the magnetic field. The calculations show that the increase of THG intensity is dominated by the magnetic field enhancement of the exciton oscillator strength, which is particularly strong for GaAs in the studied range of field strengths. The much weaker increase of THG intensity in CdTe and ZnSe is explained by the considerably larger exciton binding energies, leading to a weaker modification of their oscillator strengths by the magnetic field.