Dobbins' 52-yard score early in the third quarter opened a 24-7 lead for the Buckeyes (2-1), who were coming off a 31-16 home loss to Oklahoma on Sept. 9. Army (2-1) had a five-game winning streak snapped.
Barrett, who ran for a TD in the first half, passed 20 yards to Terry McLaurin to make it 31-7 in the third quarter. Then with 7:01 left in the game, he added a 9-yard scoring strike to Austin Mack to break former Purdue quarterback Drew Brees' conference record of 106 TDs responsible for in a career.
Dobbins rushed for 172 yards on 13 attempts.
Barrett added 32 yards on seven rushes and was 25 for 33 for 270 passing yards.
Army's Darnell Woolfolk had 74 yards on 15 runs, and quarterback Ahmad Bradshaw completed 2 of 7 passes for 19 yards.
The Buckeyes led 17-7 at the half on runs of 5 yards by Barrett and 2 yards by Dobbins in the first quarter before Army countered with a 3-yard TD run by Woolfolk in the second.
Sean Nuernberger made a 33-yard field goal late in the quarter for the 10-point margin.
OSU went 75 yards in eight plays on the opening drive, with Dobbins chewing up 36 of it on a run prior to Barrett's score with 12:23 left in the first quarter.
The Buckeyes on their next possession went 94 yards in 13 plays, with Dobbins scoring with 4:11 left in the first quarter.
Army used 18 plays and 9:37 off the clock to go 99 yards to make it 14-7 with 6:26 remaining in the half, but Nuernberger's field goal with 3:06 left put the lead to 10 at the half before OSU took over in the third quarter.