Miami (23-9) will face top-seeded Duke on Friday.
The score was tied 69-69 when BC's Makai Ashton-Langford was short on a jumper in the lane. Miami's Sam Waardenburg grabbed the rebound with 5.6 seconds left. He fed Charlie Moore, whose long outlet pass led to Miller's dramatic layup that was released a split-second before the buzzer.
Kameron McGusty led Miami with 16 points. Moore added 15 points and a game-high six assists. Isaiah Wong had 14 points and Miller tallied 13. Waardenburg added nine points and nine rebounds.
Thirteenth-seeded Boston College (13-20) had its two-game win streak snapped, with both of those victories coming in the earlier rounds of the ACC tournament.
BC was led by Quinten Post's 14 points. Brevin Galloway added 12 points.
Miami led 34-32 at halftime.
In the second half, Miami stretched its lead to 39-34 before BC went on a 6-0 run.
BC's largest lead of the game was 47-41 on Jaeden Zackery's 3-pointer with 13:50 left.
Miami regained the lead, 57-56, with 6:19 left on a Miller free throw.
Moore — despite being the smallest player on the court at 5-feet-11 — made a go-ahead put-back with 57 seconds left. Then, after McGusty made 1-of-2 free-throw attempts, Miami led 65-62. But Zackery tied the score on a 3-pointer with 15 seconds left.
Miami tried to win it in regulation, but Wong lost the ball, and Galloway's 65-foot heave just missed, bouncing off the front rim.
For the game, Miami had an 18-9 edge in points off turnovers and an 18-8 advantage on fast-break points. Miami shot 53.6 percent overall and went 6-of-19 on 3-pointers.
BC shot 50.9 percent overall, including 6-of-15 on 3-pointers.