Michigan State got the ball to start the first overtime period, but Kirk Cousins was intercepted by Bacarri Rambo on the third play.
Georgia ran the football twice before Blair Walsh tried a 42-yard field goal to win the game. The ball sailed wide right to keep the game going.
The Bulldogs started with the ball in the second OT, but lost five yards due to a false start penalty. Walsh redeemed himself by making a 47-yard field goal to give Georgia a 30-27 lead.
The Spartans responded with a 35-yard Conroy field goal to send the contest into a third overtime.
Michigan State moved the ball to the six-yard line. Faced with a 3rd-and-2, Keshawn Martin dropped a pass. Following a delay of game penalty, Conroy made a 28-yard field goal for a 33-30 edge.
A big sack by William Gholston set up a 4th-and-14 for the Bulldogs on the 29. Walsh's 47-yard field goal attempt was blocked by Anthony White to end the game.
The Spartans (11-3), who lost 42-39 to Wisconsin in the inaugural Big Ten Conference Championship Game, recorded their first bowl victory in the Mark Dantonio era. It was Michigan State's first bowl win since a victory over Fresno State in the 2001 Silicon Valley Football Classic.
"We found a way. It wasn't perfect, but we certainly showed character," said Cousins.
The Spartans quarterback completed 27-of-50 passes for 300 yards, one touchdown and three interceptions for Michigan State, which appeared in a bowl game for a school-record fifth consecutive year.
Le'Veon Bell carried the ball 17 times for 48 yards and two scores. He also caught five passes for 39 yards.
Tavarres King caught six passes for 205 yards and a touchdown for Georgia, which appeared in its 47th bowl game (26-18-3).
"We didn't execute the way that we should have at times, but credit Michigan State," Georgia head coach Mark Richt said. "We had a chance late in the game to get a first down and then the game would have been over. We hurt ourselves and didn't execute the game plan the way we should have at times. [Murray] had to throw the ball a little quicker than he wanted to, but credit MSU defense for coming out and pressuring him. Michigan State came prepared to win this game and earned every bit of the win."
Michigan State's first offensive play of the game ended in a bad way. Martin was tackled in his own end zone to give Georgia an early 2-0 edge.
The next points in the contest didn't come until late in the second quarter. Murray threw a bomb down the middle of the field for King, who caught the ball and ran into the end zone for an 80-yard score. That made it 9-0 with three minutes left.
The Spartans followed with a three-and-out on offense. Brandon Boykin took the ensuing punt 92 yards for a touchdown and a 16-0 margin. It was the longest play in the history of the Outback Bowl.
In the third, Michigan State got back in the game after Murray was intercepted by Darqueze Dennard near midfield. B.J. Cunningham and Brian Linthicum caught passes of 12 and 17 yards, respectively, to move the ball into the red zone. Bell capped the nine-play drive with an eight-yard TD run. A successful two- point conversion cut the gap to 16-8.
Dennard came up with another big pick later in the period. He returned this one 38 yards for a touchdown. A failed two-point conversion kept the score at 16-14 with under 4 1/2 minutes remaining.
Shawn Williams picked off Cousins early in the fourth, which led to a 32-yard Walsh field goal.
Linthicum took a WR screen 50 yards down the left sideline on the first play of the ensuing drive. That set up Keith Nichol's seven-yard TD catch with under 8 1/2 minutes to go in regulation. A failed two-point conversion kept Michigan State ahead by one, 20-19.
But King made a big 53-yard catch down the right sideline for Georgia. Three plays later, Boykin caught a pass over the middle of the field for a 13-yard score. A successful two-point conversion gave the Bulldogs a 27-20 edge.
The Spartans got the ball back at their own 15 with 1:55 to play. Nichol and Cunningham made catches of 15 and 22 yards, respectively, to move the ball past midfield. A 20-yard scramble by Cousins put the ball inside the red zone. Martin seemed to score with 34 seconds to go, but his knee came down at the one for an 11-yard gain. Bell rumbled into the end zone two plays later to make it 27-27 with 19 ticks left on the clock.