The Jayhawks, undaunted by the pulsating crowd, methodically built a 16-8 lead midway through the first half. Kansas took advantage of some balky outside shooting by the Wildcats, and routinely beat them off the dribble at the other end for easy baskets around the rim.
McLemore picked up his second foul of the half with 9:22 left, though, and Kansas State went on a run. Martavious Irving started the 11-2 spurt with a scooping layup and Southwell ended it with a 3-pointer from the wing that gave Kansas State a 19-18 lead.
Self called a quick timeout, settling down the senior-laden Jayhawks.
Perry Ellis scored a couple of baskets around a shot clock-beating jumper by Tharpe to help Kansas retake the lead, and Kevin Young's putback off his own miss made it 31-27 at halftime.
"It just came down to making plays at key moments," Self said. "There were several plays like that that allowed us to keep our distance."
McGruder, shadowed everywhere by Releford, only managed two points on a soaring jam just before halftime. He missed both of his outside shots as the Wildcats went 4 for 16 from the arc.
The Big 12's reigning player of the week finally hit a 3 to start the second half, but Kansas answered with eight straight points, including a dunk by Withey off a feed from Young that forced Weber to burn his first timeout of the half.
McGruder hit consecutive 3s later in the half to help keep the Wildcats close, but a 3 from the corner by McLemore and then an open jumper by the freshman with 6:50 to go gave the Jayhawks a 53-43 lead, the first time the margin had reached double figures all game.
Southwell shot the Wildcats back in it, hitting a jumper with 8:13 left and then consecutive 3-pointers — his fourth and fifth of the game — to make it 54-49 with 5:30 remaining.
But the Jayhawks managed to do just enough down the stretch to turn back the Wildcats, who won't have another chance against Kansas until they meet Feb. 11 at Allen Fieldhouse.
"We've got to start getting some wins against them," Southwell said. "We've got another chance at them, but we're focused on Iowa State."