The win means the second-seeded Cavs will now play against the winner of the Chicago Bulls-Milwaukee Bucks series, which the Bulls lead 3-1.
James grabbed 10 rebounds and dished out eight assists as the Cavs built a 21-point half-time lead and held off a Boston charge that brought the Celtics as close as eight early in the fourth quarter.
However, Cleveland guard JR Smith was ejected and they also suffered the potential loss of forward Kevin Love with a dislocated left shoulder.
Elsewhere Chris Paul scored 34 points and forward Blake Griffin added 20 along with 19 rebounds, as the Los Angeles Clippers beat San Antonio Spurs 114-105 to even their Western Conference first-round series at 2-2.
After combining for only 21 points during a heavy loss in game three, Griffin and Paul bounced back to pick up seven assists each, while guard JJ Redick supplemented the two stars with 17 points.
The team also received scoring help from reserve guards as Austin Rivers scored 16 points and Jamal Crawford 15.
Forward Kawhi Leonard led San Antonio with 26 points, forward Tim Duncan produced 22 points and 14 rebounds, and guard Tony Parker scored 18. But Duncan and Parker missed four free-throws apiece.
The Washington Wizards completed the first play-off sweep in a seven-game series in franchise history after they destroyed the Toronto Raptors 125-94.
Bradley Beal scored 23 points and centre Marcin Gortat added 21 points and 11 rebounds as the Wizards hit 15 of 26 three-point attempts.
Guard John Wall contributed 14 points and 10 assists for the Wizards, who led by at least 14 points after every quarter and by as many as 37 in the second half thanks to a three-point shooting barrage.
Seven players scored in double figures as Washington posted their highest point total for a regulation game this season and the Wizards, who are seeded fifth in the East, will face either Atlanta or Brooklyn in the next round.
The Dallas Mavericks avoided being swept by Houston with a comfortable 121-109 win over the Rockets in Game Four.
The starting Dallas backcourt of Monta Ellis and JJ Barea combined for 48 points as Ellis compiled 31 points on 13-of-21 shooting. He scored 12 points in the third quarter as the Mavs blew the game open by running out to a 24-point lead.
Houston guard James Harden scored a team-high 24 points, but made just seven of 15 shots from the floor.
The Rockets lost their way offensively after a strong start, missing 17 consecutive shots at one point, but do lead the series 3-1, with game five in Houston on Tuesday.