Barrett faked a handoff to his right, then exploded through a hole and outran Minnesota's defense. The touchdown gave the Buckeyes a 7-0 lead.
Barrett, a redshirt freshman, broke another record on Saturday. His third-quarter touchdown pass to Michael Thomas set a record for the most total touchdowns (37) by a Buckeyes quarterback in a single season.
The 36 Miller recorded last season was the previous record.
Over 10 games this season, Barrett has passed for 2,156 yards and 26 touchdowns to go with 582 yards and eight touchdowns on the ground.
Ohio State is 8-1 (5-0 Big Ten) and ranked eighth in the latest edition of the College Football Playoff rankings.