Governor Angel Aguirre told reporters in Chilpancingo, the state capital, that he was leaving office "to favour a political climate" conducive to a solution.
"The priority must be to continue with the search for the missing students and to ensure that the people responsible for these serious violations of human rights be punished," he said.
Aguirre's resignation has been one of the principal demands of the families and friends of the 43 students from the Ayotzinapa Rural Normal School, a teachers college, who disappeared Sep 26.
Protests over the mass abduction have swelled this week to include thousands of students, teachers and parents.
However, Aguirre defended his handling of the crisis and insisted that he has worked "tirelessly" to find the missing students.