Missouri (11-3) won its third straight bowl game to reach 11 victories for the fourth time in school history.
Minnesota (8-5) trailed 19-17 entering the fourth quarter, but Missouri pulled away on Hansbrough's 78-yard touchdown run and Maty Mauk's 7-yard scoring pass to Bud Sasser.
Mauk settled down to throw two TD passes after interceptions on the Tigers' first two possessions.
The Tigers forced three turnovers.
Minnesota quarterback Mitch Leidner was 21 of 31 for 258 yards and a touchdown.
The Tigers used an onside kick to open the third quarter to stretch out their halftime lead to 13-7.
The Golden Gophers responded on the ensuing series with a 54-yard touchdown pass from Leidner to tight end Maxx Williams. Williams leaped over cornerback Aarion Penton and stepped over another defender before diving into the end zone.
But Missouri quickly answered, needing just three plays after a muffed Minnesota punt for Mauk to give the Tigers a 19-14 lead on an 18-yard touchdown run.
It tightened to 19-17 by the start of in the fourth quarter before Hansbrough got free and rumbled for his score that put the Tigers' up by nine with 9:28 to play.
Missouri survived a disastrous offensive start to lead 10-7 at halftime in a game that started with the teams combining for three turnovers in the first 4 minutes.
Missouri's Mauk opened the game by being picked off by Derrick Wells on the Tigers' third play from scrimmage. Minnesota would give it right back, though, when Leidner fumbled after being sandwiched in the backfield by defensive ends Markus Golden and Shane Ray.
The Tigers then went just four more plays before Mauk's pass was intercepted for the second time, this time in the end zone by Briean Boddy-Calhoun.
The Gophers finally ended the drought on the ensuing series, capping an 11-play, 80-yard drive with a 20-yard touchdown run by Rodrick Williams.
It was part of a bruising first quarter that saw Minnesota gain 76 yards on the ground.
That was flipped in the second quarter, as the Tigers used its own rushing attack to set up the passing game.
First, following a punt, they used their best starting field position of the half to get on the board with Andrew Baggett's 21-yard field goal.
Mauk then made the best of Missouri's final possession of the half, finding Bud Sasser for a 25-yard touchdown strike.
In all, the Tigers outgained the Gophers 156-41 in the second quarter.