The Heat won the series, 4-1, and will face either Indiana or Washington in the East finals starting next week.
Miami had not led since late in the second quarter before Allen's shot from the left wing put the Heat up 93-91.
Allen made a pair of free throws with 21.6 seconds to play, capping a 12-0 run that erased an eight-point deficit.
LeBron James scored 29 points and
The The Pacers lead the series against the Wizards, 3-2, one win away from setting up a rematch with Miami that seemed like an absolute certainty for much of the season.