Five years to the day after the Celtics snapped Houston's winning streak at 22 games, James scored 37 points and made the go-ahead basket with 10.5 seconds left in Miami's 105-103 victory on Monday night.
After overcoming the largest deficit they've faced during their streak, the only team left for the Heat to catch is the 1971-72 Los Angeles Lakers, who won 33 in a row.
"It means a lot," James said. "I know the history of the game. To be sitting in second place right now ... for us to be there and doing it the way we want to do it, it means a lot."
Jeff Green scored a career-high 43 points for Boston, which was without Kevin Garnett for the second straight game. Paul Pierce had 17 points, eight rebounds and eight assists for the Celtics, but he missed a 3-pointer with 3 seconds left that could have given Boston the win.
Instead, the defending NBA champions won a regular-season game in Boston for the first time in 11 tries.
"This was a passionate game, a lot of history between the two teams," said Miami forward Shane Battier, who played for the Rockets when their 22-game win streak ended on March 18, 2008, and was with the Heat when they knocked Boston out of the playoffs in each of the last two seasons. "Not that it's playoff intensity ... nothing's like the playoffs."
James added 12 assists and Mario Chalmers scored 21 for Miami, which has not lost since Feb. 1.
"It's a special opportunity that we have with this group and you don't want to take it for granted," coach Erik Spoelstra said. "You want to treat every day as a special opportunity to be with this group, to share these moments together, but more importantly to take a step closer to going after our goal and every day that we improve puts us in a better position in a quest where nothing is guaranteed for anybody."
The Heat had not won in Boston since the New Big Three was formed -- including a double-overtime Celtics victory on Jan. 27, when Ray Allen returned with his new team.
That was the day Rajon Rondo was diagnosed with an ACL tear that ended his season.
This time, it was Garnett who sat out, though coach Doc Rivers said he would have played if the thigh injury had not been compounded by the flu.
"I believed we were going to win the game before the game. I thought we were going to win the game during the game. And I thought we should have won the game," Rivers said. "I give them credit. They made some spectacular plays."
Boston led by as many as 17 in the first half, but it was 103-all when Miami got the ball with 33 seconds left. James dribbled down the shot clock before putting up a jumper from the right wing to take the lead.
Green drove into the lane for Boston, but he was stuffed by Battier; the ball went out of bounds and, after replay, was given to the Celtics with 7.1 seconds remaining.
"No moral victories, even though it was a close game," Pierce said.
Green had 14 points in the first quarter, when the Celtics scored the last 17 points to open a 31-19 lead.
Boston led by as many as 17 in the second, but Miami went on a 15-6 run just before the half to make it a six-point game and then scored the first eight points in the third quarter to take the lead.
Green had 12 in the second quarter and 12 in the third.
"He showed what he can do," Pierce said. "That was an example of it today."
James was called for a technical foul for taunting when he stood over Jason Terry after a dunk. It was James' sixth technical of the season and the 39th of his career. ... Pierce and Chalmers got involved in a minor scuffle when Pierce drew an offensive foul and Chalmers tried to grab the ball from him after the whistle.