The Wildcats will play Wisconsin next Saturday outside of Big D.
Stauskas finished with 24 points for the Wolverines, who finished a win shy of their second straight Final Four.
Randle had 16 points and 11 rebounds for his 24th double-double and was named the region's most outstanding player.
But he was just one of the freshmen stars for the Wildcats Sunday.
While Harrison was being completely shut down early, it was unheralded Marcus Lee keeping the Wildcats in the game.
Lee, one of the six McDonald's All-American freshmen on Calipari's roster, had scored a total of nine points since the beginning of January. But he got minutes that would have normally gone to the injured Willie Cauley-Stein, and finished with 10 points and eight rebounds. Eight of those points came on put-back dunks that were part of Kentucky's 18 offensive rebounds.
Harrison's first 3 gave Kentucky a 58-55 lead and was part of an 11-0 run that put the Wolverines in catch-up mode, behind 62-55 with 6:30 left.
They fought back, and during a nine-possession stretch of sublime basketball that covered more than four minutes, each team scored every time they got the ball.
The first stop in the sequence gave the Wolverines the ball with about a minute left, trailing 72-70.
Stauskas missed a layup and a 3-pointer and Derrick Walton then missed an open 3. But the fourth attempt went in with 31 seconds left and got credited to Jordan Morgan on a scramble under the basket, though it was Randle's hand that tipped the ball in.
Calipari called a timeout. Michigan burned a foul. And the endgame started with 10 seconds left. The ball went to Harrison and it was clear he was going to take the shot. He spotted up from about 25 feet, and after he hit, he walked backward calmly before being hugged by Randle and Dakari Johnson.
Moments later, Kentucky was celebrating, preparing for the program's 16th trip to college basketball's biggest stage.