Missouri (8-2, 5-1 SEC) scored 28 points in the third quarter to erase a seven-point halftime deficit and take a 34-20 lead.
The Aggies got within a touchdown early in the fourth quarter and had a chance to tie it with less than three minutes remaining, but lost a yard on fourth down from the Missouri 1.
The Tigers then ran out the clock to secure their fourth straight win and remain in first place in the SEC East.
Hansbrough ran for touchdowns of 49 and 45 yards in his fifth career 100-yard game in front of a crowd of more than 100,000 which included former President George H.W. Bush and his wife Barbara.
Kyle Allen threw for 237 yards with three touchdowns and an interception for Texas A&M (7-4, 3-4).
Andrew Baggett's 49-yard field-goal attempt bounced off the right upright, giving the Aggies the ball back with 5 1/2 minutes remaining, but they came away empty after the fourth-down stop.
The Tigers tied it at 13-13 when Mauk found Darius White in the end zone for a 16-yard touchdown reception early in the third quarter.
A&M responded quickly with Allen finding Josh Reynolds for a 56-yard touchdown catch three plays later to put the Aggies back up 20-13. The score gave Reynolds 12 touchdown receptions this season to tie a school record. He finished with five receptions for 125 yards.
Hansbrough knotted the score once again when he burst up the middle 49 yards for a touchdown less than two minutes later.
Missouri forced a punt and Hansbrough did it again on the next drive, this time evading three tacklers on his way to a 45-yard touchdown run to make it 27-20. A 16-yard touchdown run by freshman Ish Witter pushed the lead to 34-20 with about a minute left in the quarter.
Josh Reynolds grabbed a short pass from Allen, slipped from one defender and flipped over another and into the end zone for a 24-yard touchdown to give A&M a 10-3 lead about 10 minutes before halftime.
That score came after the Aggies got a fresh set of downs when Missouri was penalized for roughing the punter.
Bud Sasser had a 2-yard touchdown catch wiped out when he was called for pass interference and the Tigers settled for a 35-yard field goal to cut the lead to 10-6. Sasser finished with 83 yards receiving.
The Tigers were driving late in the second quarter when Armani Watts intercepted a pass and returned it 36 yards. Reynolds had a 32-yard reception to set up a 31-yard field goal to leave A&M up 13-6 at halftime.
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