The Heat were down by as much as 17 points in the second quarter and trailed the Celtics by 13 at halftime before exploding for 37 points in the third frame. Miami had been 0-21 in postseason games when trailing by at least that many at the half.
The Heat are now 1-21 in such games and sit two wins away from their first NBA finals since 2014.
"We like to make it hard on ourselves," Heat star Jimmy Butler said. "We like being down double digits and being the comeback kids."
Heat sharpshooter Duncan Robinson scored 18 points on 6-of-12 shooting from 3-point range to end his offensive woes. In the Heat's last two playoff games, he was 2 of 9 from deep and scored a combined eight points.
Butler finished with 14 points, four rebounds, four steals and three assists. Jae Crowder scored 12 points, and Tyler Herro added 11 off the bench.
Kemba Walker had a team-best 23 points for the Celtics, who also got 21 apiece from Jayson Tatum and Jaylen Brown. Brown failed to make a corner 3-pointer that would have tied the game with 15 seconds left, and Butler sealed it with two free throws with 7.4 seconds left.
Game 3 of the best-of-seven series is set for Saturday.