Sophomore forward Xavier Sneed had 22 points and nine rebounds as Kansas State (25-11) prevailed in the battle of two schools nicknamed Wildcats.
Brown was just 4-of-15 shooting but knocked down the decisive one with 18.4 seconds left while scoring 13 points.
Kansas State will face 11th-seeded Loyola-Chicago on Saturday for a spot in the Final Four.
"I see a lot grit and a lot of guys who love each other," Brown said in a postgame interview on CBS. "We play together, we play defense the right way. We play for each other and we came out with the win."
Freshman forward PJ Washington contributed 18 points and 15 rebounds for Kentucky (26-11) but was just 8 of 20 from the free-throw line. Freshman guard Shai Gilgeous-Alexander tallied 15 points and freshman forward Kevin Knox added 13 points and eight rebounds.
Gilgeous-Alexander missed a tying 3-point attempt as time expired as Kentucky lost a Sweet 16 game under coach John Calipari for the first time in seven attempts.
Washington's putback off his own miss with 1:32 remaining gave Kentucky a 57-56 lead before Brown made two free throws to put Kansas State back on top.
Washington then split two free throws to knot it at 58 with 1:14 left.
Brown drove through traffic for a left-handed layup to give Kansas State the lead. Kentucky freshman guard Quade Green then air-balled a potential go-ahead 3-pointer before Kansas State junior guard Amaad Wainwright split two free throws with 7.7 seconds left to make put Kansas State up three.
Kansas State standout junior forward Dean Wade saw limited action after missing the past three games with a stress fracture in left foot. He had four points in eight minutes and did not play in the second half.
Kentucky was 23 of 37 from the free-throw line while Kansas State was 14 of 22. Kentucky shot 38.1 percent from the field and Kansas State shot 35.2 percent.
Kentucky took its first lead of the game at 36-35 on a 3-pointer by Green with 17:19 remaining.
The score was knotted at 38 when Kansas State freshman guard Cartier Diarra drained a 3-pointer to begin a 9-0 burst that saw his squad hold a 47-38 lead with 13:20 remaining.
Kentucky later used a 11-2 surge with Gilgeous-Alexander's layup giving his team a 55-54 edge with 4:02 to go.
Kansas State opened the contest with a 13-1 run and never trailed in the first half while taking a 33-29 lead into the break.
--Field Level Media
Kansas State | Kentucky | |
Xavier Sneed 22 | Scoring | PJ Washington 18 |
Barry Brown 3 | Assists | Shai Gilgeous-Alexander 5 |
Xavier Sneed 9 | Rebounds | PJ Washington 15 |
Barry Brown 4 | Free Throws Made | Shai Gilgeous-Alexander 11 |
Kamau Stokes 4 | Steals | PJ Washington 3 |
Makol Mawien 2 | Blocks | Wenyen Gabriel 1 |
Team | Points | FG% | 3PM-3PA | FTM-FTA | Assists | Rebounds | Blocks | Steals | Turnovers |
Kansas State | 61 | 35.2 | 9-22 | 14-22 | 12 | 27 | 2 | 11 | 10 |
Kentucky | 58 | 38.1 | 3-12 | 23-37 | 6 | 38 | 3 | 5 | 15 |