Jordan Clarkson scored 21 points to lead the Tigers (11-1), including two free throws with 20.1 seconds left that pushed the lead to 65-62.
Brown finished with 17, helping Missouri hold on in its first true road game this year. Johnathan Williams hit a free throw with 3.7 seconds left to make it a two-possession game and effectively seal the win.
T.J. Warren scored 24 points to lead the Wolfpack (9-3), but didn't score for the final 12 minutes as N.C. State lost despite twice leading by 10 points.
Missouri hit five 3-pointers after halftime, the last two coming in the final 4 minutes. After N.C. State's Ralston Turner knocked down one to give N.C. State a 59-57 lead, Clarkson hit one over Tyler Lewis at the 2:55 mark to put Missouri back up. 60-59.
Turner, who had made just one shot through the first 36 minutes of the game, answered with another 3 that put the Wolfpack up 62-60 with 1:39 left. But Brown responded with the shot that would put Missouri in front for good.
Missouri was coming off its first setback, a 65-64 loss against Illinois in which the Illini hit two free throws with 4.6 seconds left to spoil the Tigers' perfect start.
As for N.C. State, the loss snapped a seven-game winning streak. It also looked unlikely considering how the Wolfpack controlled the action for much of the night and didn't trail until Clarkson's three-point play with 3:51 left.
There were plenty of things for the Wolfpack to regret after this one. In addition to Warren's quiet finish, Barber had a chance to put N.C. State back in front moments after Brown's 3 only to come up short on the front end of a 1-and-1 opportunity at the foul line with 26.8 seconds left.