Blue connected on a late 3-pointer to tie the score, setting up a pair of game-clinching free throws from Jamil Wilson with 47.7 seconds remaining.
The Golden Eagles (25-8) hung on for the win when Butler's Rotnei Clarke missed a 3-point attempt from the left side with 5.5 seconds remaining.
Marquette will make its third straight Sweet 16 appearance, taking on the winner of Sunday's game between Miami-Florida and Illinois.
Clarke finished with 24 points for sixth-seeded Butler (28-9).
Notable deaths of 2014 [PHOTOS]