Murray scored all 11 of the Hawks' points in overtime, posting a career-high of 44 to help the Hawks maintain their push for an Eastern Conference play-in berth.
The Celtics, with a league-best record of 57-16, are already assured of top seed in the East, but they've dropped two games this week in Atlanta, where the Hawks erased a 30-point deficit to triumph on Monday.
The rematch was a tense back-and-forth battle that featured 41 lead changes - neither team leading by more than eight points.
Boston star Jaylen Brown's pull-up jumper put the Celtics up 122-121 with 6.6 seconds left in overtime.
Murray responded with his basket over Jrue Holiday.
"That's the best team in the league - we competed," Murray said of a Hawks team coming off a 120-106 victory over Portland on Wednesday. "We didn't use being tired, a back-to-back, as an excuse.
"We came out and competed, and these are games I want to be a part of."
Bogdan Bogdanovic scored 24 points and De'Andre Hunter added 21 points and 13 rebounds for the Hawks, who won their fourth straight game.
Jayson Tatum led Boston with 31 points and 13 rebounds but missed a potential game-winning three-pointer in the final second of regulation.
Kristaps Porzingis scored 20 points and Brown finished with 18.
But the hustling Hawks out-rebounded the Celtics 53-43 and had 28 second-chance points to Boston's 11.
"We're fighting for a play-in/play-off spot," Murray said of the Hawks team currently holding down 10th place - and the final play-in spot. "We all showed up."
In New Orleans, Zion Williamson scored 28 points and CJ McCollum added 25 to lead the Pelicans to a 107-100 victory over Giannis Antetokounmpo and the Milwaukee Bucks.
Jonas Valanciunas notched his 32nd double-double of the season with 17 points and 10 rebounds for New Orleans.
Two-time NBA Most Valuable Player Antetokounmpo scored 35 points and grabbed 14 rebounds for Milwaukee, but the Bucks dropped their second straight, unable to bounce back after a stunning overtime loss to the Los Angeles Lakers on Tuesday.
The Pelicans led by as many as 18 in the first half but the Bucks had cut the deficit to five with 3:27 left to play.
New Orleans kept attacking and closed it out from the foul line, Williamson making five free throws in the waning minutes.
"Z's been amazing, night-in, night-out," said Pelicans coach Willie Green. "He's putting the team on his shoulders."
The Pelicans pulled off the victory despite a poor shooting night. They made just 36 of their 91 attempts from the floor and just eight of 32 from three-point range.
"That was a gutsy win on our home floor," Green said. "It was a good defensive performance by us. We played with force from start to finish.
"Even when we didn't hit shots we still continued to make winning plays."
