Nebraska (18-8, 6-6 Big 12), which trailed by seven points at halftime, took a 64-53 lead on Lance Jeter's driving layup with 2:35 left. But the Cornhuskers made just six of 11 free throws in the last two minutes to help Texas get back in it.
Texas (23-4, 11-1) went on a 12-1 run, making seven of nine free throws and adding a pair of baskets. Alexis Wangmere's free throws tied it 65-65 with 1:07 remaining.
Richardson sank a pair of foul shots with 43.5 seconds remaining to put Nebraska back in front, and Drake Beranek made one of two with 24.7 seconds left.
Texas' J'Covan Brown made two of three free throws with 12.6 seconds remaining to cut the deficit to one, but his 3-point attempt at the buzzer fell short.
Nebraska hit seven of its first eight shots in the second half and took a 40-38 lead on Andre Almeida's tip-in with 16:29 remaining. It was the first time Texas had trailed in seven games.
The Huskers got back-to-back layups from Richardson, and Almeida's left-handed hook made it 50-42 with 11:39 left. Nebraska's biggest lead was 11, the last time after Jeter's layup.