Missouri had a 17-14 lead heading into the fourth quarter before the teams combined for 41 points during a wild final period.
Oklahoma State seized the lead with 5:04 to go on a 2-yard run by Desmond Roland, but the Tigers then marched 69 yards in seven plays for the decisive touchdown.
Missouri quarterback James Franklin hit Dorial Glenn-Beckham with a key 27-yard throw during that drive with the Tigers facing a third-and-9 situation at the Cowboys' 43-yard line.
Josey scored on the next play to give Missouri a 34-31 lead.
Oklahoma State moved to the Tigers' 23-yard line in the final minute, but Michael Sam forced a fumble when he sacked quarterback Clint Chelf. Shane Ray picked up the fumble and ran it 72 yards for a touchdown to clinch the victory.
Until the scoring outburst in the fourth period, both teams had struggled offensively. Chelf wound up completing 33-of-56 passes for 376 yards with two touchdowns and two interceptions. Franklin was 15-of-40 for 175 yards.
Josey's game-winning touchdown run was his third score of the game. He reached the end zone on a 3-yard run in the first period and a 25-yard run early in the fourth.
Missouri finished with a 12-2 record after reaching the Southeastern Conference title game against Auburn. Oklahoma State wound up 10-3.
The Big 12 Conference closed out a 3-3 bowl season, its biggest victory having come from Oklahoma over Alabama in the Sugar Bowl on Thursday. The Southeastern Conference is 6-2 with games remaining for Vanderbilt against Houston and Auburn against Florida State in the national title contest.