Barrett has thrown for 7,622 yards after going 14 for 22 for 275 yards. He broke Art Schlichter's mark of 7,547 set from 1978-81.
The fifth-year senior ran for 89 yards on 10 carries and passed for three scores. Running back Mike Weber returned from a hamstring injury to rush for three touchdowns.
Barrett was lifted with 3:52 left in the third in favor of redshirt freshman Dwayne Haskins, who passed for one touchdown.
Ohio State (4-1, 2-0 Big Ten) amassed more than 600 yards for the second straight game. It accumulated 631 yards after piling up 664 yards against UNLV last week.
Rutgers (1-4, 0-2) has lost 16 straight games in the Big Ten. Scarlet Knights quarterbacks Kyle Bolin, Johnathan Lewis, Hunter Hayek and Gio Rescigno were a combined 11 of 27 for just 92 yards.
The Scarlet Knights have been shut out in five of their 11 conference games since last season.
The Buckeyes cruised in the first half in the air and on the ground.
Barrett connected with Johnnie Dixon on touchdown passes of 70 and 39 yards in the second quarter, and Weber notched his third TD of the half with 46 seconds left until intermission, providing the Buckeyes with a 35-0 cushion.
Barrett was 11 of 17 for 216 yards in the first half. Ohio State outgained Rutgers, 336-77. The Buckeyes averaged 9.3 yards a play in the half.
Weber scored on the Buckeyes' second possession on a 1-yard leap and followed it with a 3-yard dive on the first play of the second quarter, giving Ohio State a 14-0 lead.
NOTES: Rutgers WR and all-purpose threat Janarion Grant did not dress because of recurring headaches. ... Ohio State DT Robert Landers returned after missing last week's game against UNLV with a lower leg ailment. ... Ohio State S Erick Smith, the second-leading tackler for the Buckeyes, did not make the trip. No reason was given. ... Ohio State defensive coordinator Greg Schiano was head coach at Rutgers for 11 seasons. ... Rutgers head coach Chris Ash spent two seasons as Ohio State's co-defensive coordinator. ... Ohio State offensive linemen C Billy Price has started 45 straight games.