Jayson Tatum scored 33 points and grabbed 11 rebounds and Jaylen Brown added 27 points for the Celtics, who will try to finish off the best-of-seven series at home on Wednesday.
The Cavaliers were dealt a heavy 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 -- their first lead since the first 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 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."
The Celtics, winners of a league-best 64 regular-season games, used a 12-0 scoring run to take charge in the first quarter, Tatum scoring 16 points in the period.
Boston led by as many as 13 in the second quarter but Tatum cooled off and Cleveland, buoyed by 11 of their 15 three-pointers in the first half, battled back, twice cutting the deficit to one point before going into the break down by five.
The Celtics caught a break early in the second quarter, when Brown was called for only a common foul, rather than a flagrant, 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 thought our guys deserved much better, the way that they were competing, they were attacking the paint, they were getting after it.
"We're not asking for anything more, but we're asking for equal and I don't think we got an equal opportunity at it tonight from that standpoint."
The Dallas Mavericks were aiming to extend their 2-1 series lead as they hosted Western Conference top seeds Oklahoma City Thunder in the later game.
