The heavily-favored Tigers trailed, 14-3, after the first quarter, but led, 17-14, heading in the the third as they out-scored the inspired Bulls, 14-0.
USF sent the contest into the extra period when Delbert Alvarado kicked an 18-yard field goal with less than a minute left in regulation.
In OT, Grothe's heroics, on a 4th-and-1 at the Tigers' 16, followed a 39-yard field goal by Auburn's Wes Byrum.
Grothe, who also ran for a score, was 18-of-27 passing for 184 yards, and Mike Ford scored once on 21 carries for 74 yards as USF improved to 2-0.
Auburn's Brandon Cox went 16-of-35 for 165 yards with a touchdown and two interceptions.
Mario Fannin rushed for a touchdown on 14 carries for 62 yards, and Ben Tate added 61 yards on 13 carries as the Tigers fell to 1-1.