The win means Erik Spoelstra's men take a 2-1 lead in the best-of-seven series and helped restore confidence off the back of a lacklustre Game 2 in which they never found their rhythm and ultimately blew a home-court advantage.
The Heat raced into a 25-point first-half lead and center Adebayo scored 31 points to offset the loss of the injured All-Star Butler at half-time because of right knee inflammation on what was a disappointing night for the Celtics.
Adebayo also had six assists - and four of the Heat's postseason record 19 steals - as the Heat survived a mini revival when their lead was cut to just one point at 93-92 with only 2:40 left to play.
"He did his version of what Jimmy does: 'Do what's necessary for the game," Heat coach Spoelstra told reporters.
''Tonight we needed the scoring and we needed that offensive punch early on.
"Then, when Jimmy was out in the second half, he just stabilized us.
"It got a little gnarly out there and when it did, we were able to get the ball to Bam and just get something coherent."
Adebayo said his team-mates had been smarting from a shock 102-127 Game 2 reverse at home and were determined to put that right.
"In the previous game, as everybody noticed, they beat us like we stole something," Adebayo said.
"That should wake everybody up. Getting beat at home like that, that says it all."
Now at worst they can head back to Miami locked at 2-2 and Kyle Lowry, who was back from the sidelines following a hamstring injury he picked up in Game 4 of the Eastern Conference semi-finals against the Philadelphia 76ers, said the Heat had put themselves back in contention.
"No matter what happens, at the end of the day, if you put your heart out there and play hard, you've got to be happy and just live with the results," Lowry said. "And that's what we do."
Spoelstra revealed Butler would not need another MRI on the knee but he is a major doubt for Game 4 in Boston on Monday.
"He didn't have his, like, normal explosive burst," the coach said. "He's been able to manage this. I think the next two days will be really important, obviously."