Zydrunas Ilgauskas contributed 21 points and 10 rebounds for the Cavaliers, who also had 18 points and six assists from Larry Hughes in winning for the 13th time in their past 16 games.
"It definitely means something," James said of beating the Celtics. "They're number one in every (defensive) category and we did a good job offensively executing. It was two good teams and we slugged it out to the (end)."
The Celtics, who fell to 36-9, led 66-64 at half-time, scoring the most points allowed all season by Cleveland in the opening half.
Shaq goes to Phoenix
"You shoot 57 percent from the field and score 113 and lose. That's a hard one to swallow," Celtics coach Doc Rivers said. "So many things defensively (broke down). Our offence was bailing us out enough to stay in the game."
The Cavaliers led 112-110 thanks to a dunk by Ira Newble, the last of his 15 points, when Daniel Gibson stole the ball from Boston's Rajon Rondo with 40 seconds to play and grabbed a rebound to set up two key Ilgauskas free throws.
"I saw (Rondo) coming and I just wanted to try to make a play," Gibson said. "I hope he wasn't asking for a foul because I got a lot of ball."
The Cavaliers led by eight points before the Celtics pulled within 110-108 on two Rondo free throws with 1:48 to go, setting the stage for the final dramatics.
"We've got to find ways to win games because we dug ourselves a hole early in the year," Cavaliers coach Mike Brown said. "Neither team could really stop the other team defensively and it turned into a shootout."
Cleveland's Drew Gooden suffered a right groin injury on a spectacular rebound dunk early in the third quarter and was pulled from the game for the remainder of the night.
Newble took an accidental elbow to the face under his right eye from Ilgauskas and four stitches were required to close the gash.
Boston's Ray Allen scored 20 of his 24 points in the first half, when the Celtics sank 61.5 percent of shots from the field.
"We've been tremendous on the defensive end and for our standards, they scored too many points," Allen said. "They pretty much had their way with us. A lot of times we were out of position and it was all of us."
Rando scored 20 points and Paul Pierce added 19 for the Celtics, who played without star Kevin Garnett, idled by an abdominal strain.
Pau Gasol, meanwhile, marked a successful debut for the Los Angeles Lakers by scoring 24 points and grabbing 12 rebounds in a 105-90 road victory over the New Jersey Nets.
Acquired from the Memphis Grizzlies last week, the Spanish forward-center sat out Sunday's contest because of a sore back but enjoyed a stellar first game on a night when Lakers All-Star Kobe Bryant managed just six points.
"I felt pretty comfortable," Gasol told reporters.
"I still don't have a lot of the wrinkles and details. It always makes the game much easier when five guys on the court know what they are doing."
Lakers coach Phil Jackson said he was "not surprised" at Gasol's output.
"He's a skilled player," the coach said of the seven-footer.
The 24 points by Gasol were the most scored by a debuting Lakers player since Magic Johnson had 26 in his first game as a rookie in 1979.
Bryant, playing the last three quarters with a dislocated right little finger, was 3-for-13 from the field and was held to single digits for the first time in over a year.
He also had seven turnovers while contributing eight assists and five rebounds and working well with Gasol, who also had four assists.
Derek Fisher weighed in with a season-high 28 points to help overcome the tough offensive night recorded by Bryant.
Vince Carter scored 27 points and Bostjan Nachbar had 19 points and 10 rebounds for the Nets.
Tim Duncan had 19 points and 15 rebounds to lead the San Antonio Spurs to a 116-89 road triumph over the Indians Pacers, who lost their seventh straight game.
The Philadelphia 76ers hit the visiting Washington Wizards with a 17-0 run in the fourth quarter to pull away for a 101-96 victory.
Mo Williams had 32 points and seven assists to lead the Milwaukee Bucks to a 102-97 road win over the Memphis Grizzlies.
Round-up: Jazz on fire
Round-up: Rampant Rasheed