Barry Brown Jr.’s layup with 19 seconds remaining put Kansas State ahead to stay at 60-58, and Amaad Wainwright added 1 of 2 free throws for the final margin.
Kentucky had a final chance to tie, but Shai Gilgeous-Alexander’s clean look at a three-pointer clanked off the iron at the final buzzer.
Kansas State (25-11) faces upstart Loyola Chicago, a No. 11 seed, on Saturday for the right to advance to the Final Four.
Kentucky (26-11) took the lead for only the second time in the game on Gilgeous-Alexander’s layup with 4:04 left, but Kansas State forced a turnover that resulted in Cartier Diarra’s layup while drawing a foul. The redshirt freshman missed the bonus free throw, leaving Kansas State in front 56-55.
Kansas State bumped its lead to eight with 9:52 left in the second half thanks to a five-point run capped by Xavier Sneed’s layup, but Kentucky clawed back within 54-52 on the strength of 1 of 2 free throws by P.J. Washington, Hamidou Diallo’s layup, two more Washington foul shots and Kevin Knox’s three-pointer that brought Kentucky fans out of their seats.
Washington then sank 1 of 2 free throws with 4:43 left in regulation, and Kentucky trailed by a single point.
Trailing for the majority of the game and by four at halftime, Kentucky scored five in a row to take its first lead at 36-35 with 17:21 remaining in regulation. Those points came courtesy of two free throws from Knox and Quade Green’s three-pointer.
But Kansas State forced a pair of turnovers, one by Green and the other by Gilgeous-Alexander, and reclaimed the lead on Sneed’s three-pointer a minute and a half later. It was Sneed’s fourth three-pointer of the game and the eighth overall for Kansas State.
It used a 9-0 burst to grab a 47-38 lead, going 4 for 4 from the field in that time, including Brown’s layup with 13:17 to play that led immediately to a timeout from Kentucky Coach John Calipari, who clearly was not pleased with his team’s four turnovers over four minutes in the second half.
Kentucky ended the drought with Knox’s jumper and with Calipari jumping in front of the bench to catch a glimpse of the proceedings that had the highest seed left in the South Region teetering.
The storied program has won eight national championships, most recently in 2012 under Calipari, and made its 26th appearance in the regional semifinals. It was Kentucky’s seventh berth to the round of 16 under Calipari, more than any other team since 2010.
Kentucky, which outlasted Davidson, 78-73, in the round of 64 and clobbered Buffalo, 95-75, two days later, entered 6-0 in the round of 16 under Calipari and winners of eight straight overall in the regional semifinals.
Kansas State, meanwhile, advanced to the round of 16 for the first time since 2010 by beating Creighton, 69-59, in its NCAA tournament opener and ousting Maryland Baltimore County, 50-43, ending the improbable run of the first No. 16 seed to beat a No. 1. The Retrievers dismantled Virginia, 74-54, last week in the round of 64 in Charlotte.