Jayson Tatum scored 33 points and grabbed 11 rebounds and Jaylen Brown added 27 points for the Celtics, who beat the Cavaliers 109-102 for a 3-1 lead in their best-of-seven Eastern Conference semi-final.
The Celtics will try to finish off the series at home on Wednesday.
Western Conference top seeds Oklahoma City head home tied 2-2 with the Mavericks after a furious fourth-quarter rally carried them to a 100-96 victory in Dallas.
Oklahoma City trailed most of the night in the face of a stout Dallas defensive effort that included 13 blocked shots.
But the Thunder broke through in the fourth quarter, tying it at 86-86 on Shai Gilgeous-Alexander's fadeaway jump shot with 4:02 to play.
Rookie Chet Holmgren followed with a three-pointer that gave the Thunder the lead for good.
Dallas had the deficit down to one point with 10.1 seconds left, but Holmgren and Gilgeous-Alexander each made a pair of free throws and the Thunder closed it out.
"We just stuck to it," said Gilgeous-Alexander, who scored 22 of his 34 points in the second half.
"We just plugged away, took it possession by possession and eventually the game turned for us."
Holmgren finished with 18 points and Luguentz Dort had 17 for the Thunder, who made 23 of their 24 free-throws and withstood a triple double of 18 points, 12 rebounds and 10 assists from Dallas star Luka Doncic.
P.J. Washington led Dallas with 21 points but star guard Kyrie Irving was held to nine.
Doncic said it wasn't a defensive breakdown that cost the Mavs but too many mistakes in the "little details."
He called it "unacceptable" that Dallas made just 12 of their 23 free-throws, and the Mavericks also coughed up 14 turnovers leading to 19 Thunder points.
'A game of runs'
In Cleveland, the Cavaliers were dealt a blow when Donovan Mitchell, who had averaged more than 35 points over the six prior games, was ruled out with a calf injury, joining starting center Jarrett Allen on the sidelines.
NBA superstar LeBron James, who led the Cavs to their only NBA title back in 2016, was sitting courtside, but with Mitchell absent the Cavs ultimately didn't have enough firepower.
The Cavs kept the pressure on, taking the lead briefly on Darius Garland's driving basket early in the third quarter.
But the Celtics quickly reasserted themselves and led by 10 going into the final period.
Cleveland, on the back of 30 points from Garland, pulled within five points three times in the final four minutes, but Brown, fed by Tatum, connected on a three-pointer with 1:09 to play that effectively sealed it.
"It's a game of runs," Tatum said after the Celtics - who took control early with a 12-0 scoring run in the first quarter - struggled to put the depleted Cavaliers away.
"It's not going to be perfect every single time. They're going to make shots, but it's our job to figure it out."
Boston led by as many as 13 in the second quarter but Cleveland, with 11 of their 15 three-pointers in the first half, twice cut the deficit to one point before going into the break down by five.
Brown avoided a flagrant foul call in the second quarter after he fell backwards into Max Strus and grabbed Strus's ankle as the Cavs player - himself struggling to stay upright - stepped over his head.
Cleveland coach J.B. Bickerstaff was more concerned at the fact that Boston went to the free-throw line 24 times compared to Cleveland's seven.
"I'll be honest with you, I was disappointed with the way the whistle blew tonight," he said. "I don't think we got an equal opportunity at it tonight from that standpoint.
Garland called the free-throw discrepancy "ridiculous."
"I'm not one of those guys with the striped shirt, but I know how many times I get hit, I know how many times my teammates get hit, put on the floor. And we can't reciprocate," he said.
