TCU (7-6) rallied to set up Jaden Overkrom for a 53-yard field goal with 2:42 left, but left Conroy too much time for his second straight postseason game-winner after beating Georgia with a 28-yard kick in the third overtime of last year's Outback Bowl.
Trevone Boykin threw for 201 yards on 13-of-29 passing with an interception for the Horned Frogs.
TCU and Michigan State came to the desert with an awful lot of similarities.
The Horned Frogs opened their first season in the Big 12 with four straight wins before losing four of their final six games. Michigan State started 4-2, then lost four of six down the stretch.
Michigan State had the nation's fourth-best defense and was 10th in scoring defense during the regular season. TCU was 18th in total defense and 10th against the run.
The biggest difference between the teams was Bell.
The junior was third nationally with 137.3 yards rushing per game and had 1,648 on the season, second-most in Michigan State history and 242 fewer than TCU had as a team.
Early on, the Horned Frogs gave him nowhere to go.
Filling holes inside and stringing plays out toward the sidelines, TCU stuffed the 237-pound junior on nearly every touch, holding him to 38 yards on 11 carries in the first half.
Of course, it didn't seem to matter what Michigan State did. The Spartans had 29 yards on 12 plays in the first quarter and struggled in the second.
Michigan State had three first downs and 76 total yards in the first half.
The Spartans still seemed to be stuck in the ruts in the third quarter before grinding out a 14-play scoring drive led by Cook, who replaced Maxwell for the second time in the game. They had their biggest play on a floating pass from Bell to fullback Tyquan Hammock (29 yards), then Cook threw his first career touchdown pass, a 15-yarder to Burbridge on a crossing route that cut TCU's lead to 13-7.
TCU didn't exactly have its way with Michigan State's defense, but did enough to put together three first-half scoring drives.
The Frogs started gashing the Spartans for decent-sized chunks with their option midway through the first quarter, setting up Matthew Tucker's 4-yard touchdown on an end-around.
Boykin an off-the-back-foot throw to freshman Kolby Listenbee for 59 yards on the last play of the first quarter and nearly had a 19-yard touchdown pass to open the second, but Ladarius Brown was bumped and dropped the ball in the end zone. Oberkrom followed with a 47-yard field goal and added another from 31 yards after Boykin hit Josh Boyce on a 61-yard pass to put TCU up 13-0 at halftime.
The Frogs couldn't keep it up in the second half.
TCU had 30 yards of offense in the third quarter and continued to labor in the fourth before its last-ditch drive ended in the closing seconds.