The female teacher, who is in her 40s, confessed to stabbing the student in the central city of Daejeon, police said.
The girl was found with stab wounds on the second floor of a school building at 18:00 local time (09:00 GMT) on Monday and was pronounced dead at hospital. The teacher was found beside her with stab wounds that police said may have been self-inflicted.
The Daejeon education office earlier said the teacher had requested a six-month leave of absence citing depression on 9 December, but had returned to school just 20 days later after a doctor assessed her as being fit to work.