He finished third in the Heisman Trophy balloting and set 28 Ohio State records plus seven Big Ten Conference records, including single-season passing yards (4,831), touchdown passes (50) and total offensive yards (4,939).
Along the way, he torched Michigan's top-ranked defense for 396 passing yards and six touchdowns then went for 499 yards and five more touchdowns through the air a week later as the Buckeyes beat Northwestern in the Big Ten championship game.
The last Ohio State quarterback drafted was Cardale Jones, who went to the Bills in the fourth round in 2016.
Haskins is the second Ohio State player taken this year.
The San Francisco 49ers chose defensive end Nick Bosa with the No. 2 pick.