We measure the gate voltage (V_{g}) dependence of the superconducting properties and the spin-orbit interaction in the (111)-oriented LaAlO_{3}/SrTiO_{3} interface. Superconductivity is observed in a dome-shaped region in the carrier density-temperature phase diagram with the maxima of superconducting transition temperature T_{c} and the upper critical fields lying at the same V_{g}. The spin-orbit interaction determined from the superconducting parameters and confirmed by weak-antilocalization measurements follows the same gate voltage dependence as T_{c}. The correlation between the superconductivity and spin-orbit interaction as well as the enhancement of the parallel upper critical field, well beyond the Chandrasekhar-Clogston limit, suggest that superconductivity and the spin-orbit interaction are linked in a nontrivial fashion. We propose possible scenarios to explain this unconventional behavior.