Miami were rocked by the withdrawal of Butler shortly before tip-off in Florida as they prepared to face the Eastern Conference leaders and most bookmakers' favorites for this year's NBA crown.
But Adebayo stepped up with a 30-point performance as Miami sent Boston spinning to a second straight loss following Monday's reverse to Orlando.
Adebayo's haul included a crucial contribution down the stretch as Miami overturned a 10-point deficit with just over eight minutes remaining to close out a hard-fought victory.
The Celtics looked set for morale-boosting road win as they opened up an 87-77 lead midway through the fourth quarter.
But back-to-back three-pointers from Haywood Highsmith got Miami to within four points.
Adebayo then delivered a hook shot and a dunk in quick succession to make it 87-87 and then scored the go-ahead basket to give Miami a lead they would not relinquish as they closed out the win.
"Everybody has to take a bigger responsibility, everybody's got more added to their plate," Adebayo said afterward.
"This team is built for that. We've had guys in and out of the rotation, injured or not. Just having that willingness to win, that's all I can ask for from my team-mates."
Adebayo's performance was the sort of display that could put him into All-Star Game contention as the NBA's mid-season break looms.
The towering Miami center however played down speculation that he could earn a second All-Star call-up after his lone appearance in 2020.
"I'll just let the chips fall where they land, man," Adebayo told TNT television. "I'm out here hooping, enjoying this atmosphere, enjoying my team.
"All-Star game? I'd love to be in it, but it's up to the coaches. But they've seen my body of work so far."
Miami's other main scoring contributions on Tuesday came from Highsmith, with 15 points off the bench, Max Strus (13 points) and Victor Oladipo (12).
Boston's scoring was led by Jayson Tatum with 31 points while Derrick White added 23 points.
The Celtics welcomed back Robert Williams from injury but were missing defensive stalwart Marcus Smart, Al Horford and Malcolm Brogdon.
Miami improved to 27-22 with the win and are sixth in the Eastern Conference.
Elsewhere on Tuesday, Nikola Jokic returned from a two-game injury absence to lead the Denver Nuggets to a nail-biting 99-98 win over the New Orleans Pelicans.
Jokic drained a jump shot with 16.9 seconds remaining to give Denver a one-point lead -- and victory -- after a battling Pelicans fightback in the fourth quarter.
At Madison Square Garden meanwhile, a 36-point performance from Julius Randle helped the New York Knicks to a 105-103 win over the Cleveland Cavaliers.
Donovan Mitchell and Jarrett Allen scored 24 points each for the Cavs, but the Knicks dug in for a win that keeps them firmly in the Eastern Conference playoff hunt, one win adrift of Miami.