Jayson Tatum inspired the Celtics with 33 points, 11 rebounds and assists, while five other players reached double figures in what was a determined team display from Joe Mazzulla's side.
After the Heat's 128-102 win on Sunday, the Celtics were facing the possibility of a humiliating 4-0 exit but they will now be looking to put further pressure on Miami in Thursday's clash back in Boston.
While no team in NBA history has ever come back from 3-0 down to win a best-of-seven playoff series, Boston will have drawn belief that they can now deliver a comeback and reach the Finals where the Denver Nuggets await.
"We weren't thinking about winning four in a row tonight, we were just trying to save our season and win tonight and we did that," said Tatum.
Jaylen Brown, who put up 16 points, said the Celtics had faced up to their predicament and united ahead of the game.
"We wanted to make sure that we stayed together, we looked each other in the eye and we came out together and I am proud of the group. You see (some) teams, with their backs against the wall, they just collapse, we didn't do that, we came together. I think that showed our character," he said.
The Heat, looking to reach the NBA finals for the seventh time as they search for their first NBA Championship since 2013, started strongly and took a 22-15 lead with four minutes left in the first quarter.
With Gabe Vincent in inspired form, the Miami crowd sensed another convincing win for Erik Spoelstra but Boston kept themselves in the game and went in just 56-50 down at half-time.
But Miami never recovered from allowing a rapid 18-0 Celtics run in the third quarter with Tatum claiming half of those points as the visitors went from nine points down to nine up.
Miami ran out of ideas against a strong Boston defence, who had 16 steals, and the Celtics never looked in danger from then on.
Jimmy Butler put up 29 points for Miami but managed only one of four three-pointers and his team made only eight-of-32 attempts from that range.
"I think it's probably pretty easy to keep our three-point shooting down," said Spoelstra.
"We just weren't doing things with a lot of oomph to it offensively".
Boston in contrast made 19-of-45 three-pointers. The Celtics are 30-6 when they make 18 or more three-pointers in a game.
Butler, who has led eighth-seeded Miami's unexpected playoff surge, said they wouldn't allow the Celtics to gain momentum from the win.
"No, if anything it will build momentum for us, knowing that we have to play with a lot more energy. Knowing that we have to play with our backs against the wall. I think all year long we've been better when we had to do things our way," he said.
Butler said he wouldn't be taking a different approach in the wake of the loss.
"(I'll) do the same thing that I do after every game. Listen to some music, drink some beers back there go and have some wine. I don't think that you can just focus on basketball. You have to be able to get away from the game. We're going to smile and be in this thing together like we always are and go and win one on the road," he said.
Vincent, who made 17 points, suffered an ankle injury in the fourth quarter and went off for treatment before returning but Spoelstra said it was not clear how serious the injury was.
Miami might not be panicking but Boston certainly head home with some renewed belief.
"We've just got to go win another one. That's all that matters," said guard Marcus Smart.
"We understand the odds are stacked against us, but we're a team that believes in ourselves no matter what"
