Johnson's key play came with 23.3 seconds left and gave second-seeded Jayhawks (29-6) a 61-60 advantage.
Kansas trailed by as many as 10 points in the second half and didn't take its first lead until Johnson's 3-pointer with 3 minutes remaining.
He finished with 18 points while Travis Releford and Tyshawn Taylor contributed 10 apiece for the Jayhawks, who now move on to play North Carolina State for a spot in the Elite Eight.
Hummel finished with 26 points and nine rebounds the 10th-seeded Boilermakers (22-13).
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