After Kyle Federico missed a 43-yard field goal on the final play of regulation for the Scarlet Knights, Carr capitalized right away by lobbing a 25-yard pass to Greg Watson to give the Bulldogs the lead.
Gary Nova answered with his fifth TD pass of the game to Brandon Coleman. But coach Kyle Flood opted to go for the win and Nova's pass tipped off Coleman's finger tips to end the game.
Carr completed a Mountain West and school-record 53 of 74 passes for 470 yards to start his senior season off with a win. He focused mostly on three receivers with Davante Adams and Josh Harper each catching 14 passes and Isaiah Burse adding 13.
Nova threw for 348 yards, Paul James ran for 182 yards and freshman Janarion Grant returned a kick 100 yards for a score the first time he touched the ball in college but it wasn't enough to beat Carr and the Bulldogs.
It's been quite an emotional week for Carr, whose newborn son Dallas finally got to come home after spending his first three weeks in neonatal intensive care because of digestive tract defects. Carr spent much of fall camp shuttling between the practice field and the hospital, not telling teammates of his family's ordeal until detailing the difficult past month this week in an interview with The New York Times.
The game went to overtime only after Kyle Federico missed a 43-yard field goal wide right on the final play of a roller-coaster fourth quarter that featured four touchdown passes, one made field goal and a sequence of three turnovers in three plays.
Nova threw two TD passes, including a go-ahead 30-yarder to Leonte Carroo with 1:18 remaining. But Carr answered that with his second touchdown pass of the quarter when he threw a 2-yarder to Justin Harper with 38 seconds left.
Instead of playing for overtime, Rutgers came out throwing and Nova completed four passes to move the ball to the 26 to set up a potential winning kick. But after making his first three kicks of the game, Federico missed at a crucial time.
Rutgers used two long runs by James in the third quarter to set up a pair of scores that gave the Scarlet Knights a 31-28 lead heading into the fourth quarter.
Fresno State started fast by taking the opening kickoff and driving 74 yards for a score when Josh Quezada ran it in on fourth-and-1 from the 3.
Rutgers responded with 20 straight points with Grant's kickoff return and a 69-yard TD pass from Nova to Leonte Carroo.
The Scarlet Knights looked ready to take control when Carr appeared to lose a fumble on a shotgun snap in his own territory. But the call was overturned on replay because Carr briefly had possession with his knee down.
The Bulldogs took advantage of the call when Grant muffed a punt at his own 41, setting up Malique Micenheimer's 1-yard run. Fresno State took a 21-20 lead when Carr found Adams for a 2-yard score with 1 second left in the half.