With the score locked at 3-3, the series now moves on to a winner-takes-all game seven clash, which will take place in Miami on Sunday night (Monday 01.30 BST).
The decider was forced after Jimmy Butler put on a career-best play-off performance to keep the Heat in contention courtesy of their 111-103 win.
With two minutes remaining, and the scores tied at 99-99, Butler's three-pointer and P.J. Tucker's successful free throw attempts put the visitors in control.
Butler, who scored just 27 points over his last three appearances, was simply unstoppable as he made 16 of 29 field-goal attempts, including four three-pointers and 11 successful free-throw attempts.
The 32-year-old ended the night with his second 40-point game of the series and his fourth of the ongoing play-offs.
Speaking after the match, he said: "We know what we're capable of. We get to take it back to the crib and win game seven.
"We knew we were going to come here and get one. We know we're going to win this next one too."
Butler, who battled on despite a sore right knee, also thanked former Miami guard Dwayne Wade for inspiration, saying: "He was telling me I could do this. He said 'go out there and continue to build your legacy' and it meant the world to me."
Miami Heat last won an NBA championship back in 2013, and last reached the finals in 2020 when they suffered defeat against the Los Angeles Lakers.
Boston Celtics, on the other hand, claimed their last NBA crown in 2008 while their last final appearance was in 2010.
The winner of Sunday's decider will take on the Golden State Warriors, who defeated the Dallas Mavericks in the Western Conference finals to book their spot in the NBA finale.