Jimmy Butler starred, scoring 28 points, seven rebounds and six assists, as the Heat, who led the seven-game series 3-0 before being pegged back to 3-3, reached this year's showpiece.
The first eighth seeds to reach the finals since the New York Knicks in 1999, the Heat will play the the Denver Nuggets in the best-of-seven finals.
Game one will be played in Denver on Thursday (01:30 BST Friday).
Derrick White's dramatic buzzer-beater in game six had tied the series, giving the The Celtics hope of becoming the first side in NBA history to overturn a 3-0 deficit in a play-off final.
But alongside Butler's efforts, Caleb Martin scored 26 points and Bam Adebayo recorded 12 points, 10 rebounds and seven assists as the Heat silenced a passionate Boston crowd.
Butler, 33, was named MVP of the series after averaging 24.7 points, 7.6 rebounds and 6.1 assists.
"We stayed together as a group," he said.
"We talked about going to get a tough one on the road and we did just that. [My team-mates] are the reason we're here. I know how good a team we are and we made it happen."
Heat coach Erik Spoelstra said he never doubted his squad's resilience.
"Sometimes you have to suffer for the things you really want," he said.
"This group has shown fortitude when there are inevitable let-downs and failures, but had that perseverance to pick yourself up, that collective spirit to keep on forging ahead."
The Heat are seeking a fourth NBA title and first since 2013, while The Nuggets, who beat the Los Angeles Lakers in the Western Conference play-offs, are in search of a maiden championship.