Kyle Singler added 13 points for the Blue Devils (32-4), who shot 51 percent, never trailed in the second half and advanced to the round of 16 for the 12th time in 14 years.
Darius Morris scored 16 points to lead the eighth-seeded Wolverines (21-14), who trailed by 15 with 10:51 to play but clawed within one point twice in the final 90 seconds. They had one last chance after Smith missed a free throw with 8.7 seconds left.
Morris' runner in the lane with 2 seconds left bounced off the back iron and Smith grabbed the rebound at the buzzer, sealing Duke's trip to Anaheim, Calif., for the West regional semifinals Thursday night.
Krzyzewski improved to 900-283 in his 36th season, and can catch his mentor and college coach for first on the career list next weekend. He would match Knight with a victory in the regional final, and would pass him with one win in Houston that also would put the reigning national champions back in the title game.
Still, Michigan nearly made Krzyzewski wait until next year - or at least work overtime - to try for No. 900.
Coach John Beilein frustrated Duke by mixing his defenses, and Tim Hardaway Jr. reeled off seven consecutive points down the stretch - capped by a 3-pointer that made it 70-69 with 1:27 remaining. Duke milked the clock before Irving's banked-in jumper with 32 seconds left, but Morris followed that with a layup that cut it to 72-71 with 10 seconds left.
Duke fouled Smith with 8.7 seconds left, and he made his first attempt but missed the second. Zack Novak got the rebound and got it to Morris, who headed straight for the lane and put up Michigan's last shot. After the horn sounded, Morris laid face-down on the floor and repeatedly slapped the court with his palm in frustration.
Hardaway scored 15 points, Evan Smotrycz had 13, Novak scored 12 and Jordan Morgan finished with 10 for the Wolverines, who don't have a senior on the roster but were denied their first trip to the round of 16 since 1994.
This was the first postseason meeting between the two big-name schools since the 1992 national championship game, won by Christian Laettner and the Blue Devils over the Wolverines and the Fab Five.
The Blue Devils slogged through a largely uninspired afternoon and allowed the scrappy Wolverines to hang around.
And with 16:59 to play, Krzyzewski had seen enough.
After Smith missed a layup and the rebound went out of bounds, Coach K called a timeout and could be seen zipping from side to side in the Duke huddle. At some point during the break, Krzyzewski took off his suit coat - a clear sign that he meant business - and coached the rest of the game without it.
That sparked the 19-6 run that gave Duke its largest lead. Smith, the Atlantic Coast Conference's player of the year, scored 10 straight points to start the burst. Singler made it 58-43 with a three point play with roughly 11 minutes to go to put the Blue Devils in control - or so it seemed.