Tony Allen scored a career-high 23 points but was 3-of-11 from the free-throw line for Boston, which lost its fifth straight game.
Cleveland (49-32) closes the regular season at home Wednesday against Atlanta and has a chance to win 50 games for the first time since 1992-93. Even though the Cavaliers are closing in on that mark, they are locked into the fourth seed in the Eastern Conference and are being cautious not to risk injury with the playoffs beginning this weekend.
"At this point, not a lot of guys in their career win 50 games, so that's always a plus," Hughes said. "But staying healthy is definitely more important. It would be nice to get it done, but you don't want to sacrifice what we've worked all season for."
Hughes, who missed 45 games with a fracture in his right middle finger before returning to the lineup April 4, left the game midway through the first quarter after banging the right hand while in traffic. X-rays were negative, and he returned in the second quarter.
Cleveland forward Drew Gooden left the game early in the third quarter with a left groin strain. He did not return and is listed as day-to-day.
"If we really needed him to, he could have gone back in the game," Cleveland coach Mike Brown said. "It's a Catch-22 because you've got to play your guys. But you just hope you don't have to play anybody over 35 minutes a game in this situation."
Both teams were playing without their best players. James was a late scratch with a sprained left ankle, while Boston forward Paul Pierce missed his second consecutive game with a sore left elbow.
Even without James, who is expected to be ready for the playoffs, the Cavaliers had little trouble on offense. Cleveland led 82-63 after three quarters before most of its starters sat for much of the fourth quarter.
Flip Murray replaced James in the starting lineup and scored 11 points for Cleveland, which has won four of six. The Cavaliers improved to 2-0 without James in the lineup.
The Celtics, on the other hand, struggled for most of the game without Pierce.
Gerald Green, making his second career start, scored 17 for the Celtics in place of Pierce.
Greene scored a career-high 14 points and Ryan Gomes added 13 for the Celtics, whose 22 turnovers led to 30 Cleveland points.
"As a group, the young guys are getting better and better with each game," Allen said. "Tonight we had a good opportunity, but we came up with a loss."
Varejao had 12 points in the third quarter to help Cleveland outscore Boston 34-14 after trailing 49-48 at halftime.
After the Celtics pulled to 56-55 on Gomes' jump shot with 8:56 remaining in the third quarter, Varejao scored six points to spark a 19-0 Cavaliers run. Hughes recorded two points, four assists and two steals during that spurt.
"In the second half, I thought our guys came out and they played basketball the right way," Brown said. "We were able to get stops and we were able to get out on a run. And that lasted for most of the third quarter."
Boston chipped away at the lead and pulled within 91-88 on Orien Green's layup with 56.7 seconds remaining. But Donyell Marshall made a pair of free throws with 15.2 seconds left to seal the win.
"It was a good fourth quarter," Boston coach Doc Rivers said. "A good learning tool."
Dwayne Jones recorded a career-high 13 rebounds for Boston. ... Celtics point guard Delonte West was out with a strained left calf. ... Cleveland forward Drew Gooden left the game early in the third quarter with a left groin strain. ... Boston outrebounded Cleveland 36-29.