Jaylen Brown and Jayson Tatum led the Celtics fightback as they avoided a third loss on the bounce in Florida, BBC Sport reported.
It was just the second loss in 12 playoff games for the Heat.
"We played some good basketball all the way through," said shooting guard Brown, who bagged 26 points while Tatum landed 25 points and 14 rebounds.
The Celtics also welcomed back star forward Gordon Hayward from injury and he managed six points in a 31-minute spell as he was eased back into action following the ankle problem that had sidelined him since August.
Celtics coach Brad Stevens said Hayward "looked pretty good" as his side controlled the game from the start, taking a 13-point cushion into half-time and leading by 20 at one point in the second half.
Hayward said his ankle was "pretty sore" after the game but that he was "proud to get the win."
Despite Bam Adebayo's contribution of 27 points and 16 rebounds and six players hitting double figures, the Heat were outplayed and a late fightback to within three points was snuffed out.
"They came out with great force in this game, and you do have to credit them for that," Heat coach Erik Spoelstra said.
The fourth game in the series takes place on Wednesday.
In the Western Conference, the LA Lakers will aim to go 2-0 up in their finals series with the Denver Nuggets today.