The suspected gunman, who acted alone, was arrested after he was chased out of Chardon High School by a teacher. Police said the teen turned himself in to two bystanders on the street.
The slain student was identified as Daniel Parmentor.
His parents said their son's death is a senseless tragedy.
"Danny was a bright young boy who had a bright future ahead of him. The family is torn by this loss. We ask that you respect our privacy during this difficult time," they said in a statement.
Two students, identified by their friends as Russell King and Demetrius Hewlin, were hospitalized in critical condition at MetroHealth Medical Center in Cleveland, The (Cleveland) Plain Dealer reported. Joy Rickers, 18, and Nick Walczak, 17, were hospitalized at Hillcrest Hospital with less serious injuries, the newspaper said.
Police did not release the name of the alleged gunman Monday afternoon but The Plain Dealer identified the teen as T.J. Lane and said Lane was in the cafeteria waiting for a bus to take him to a school in Willoughby that offers vocational skills training for at-risk students.
High school junior Nate Mueller said he was grazed by a bullet above his right ear when the shooter opened fire in the cafeteria. Mueller said he was sitting at his regular table with his friends, waiting for a bus to take him to culinary arts classes at Auburn Career Center in Painsville.
"My friends were crawling on the floor, and one of my friends was bent over the table, and he was shot," Nate told The Plain Dealer. "It was almost like a firecracker went off. I turned around and saw [T.J] standing with a gun and I saw him take a shot."
CNN said police recovered a handgun and gave it federal investigators for examination.
Chardon is located 35 miles east of Cleveland.
Chardon school officials said grief counselors are being made available to students, school employees and parents. There will be no classes on Tuesday, newspaper reported.
"There's not a lot you can say when you are responsible for 3,100 kids," Superintendent Joe Bergant said at a news conference Monday afternoon outside the Board of Education. "I want people to stay home tomorrow to reflect on their families and if you haven't hugged or kissed your kid, do."