Zajac also had a power-play goal in the second period, and Ilya Kovalchuk added his 25th goal -- also on a power play -- for New Jersey. The Devils have won five of six and are 21-3-2 since Jan. 8.
Blake Wheeler and Dustin Byfuglien scored for the Thrashers. Atlanta, which has just three regulation wins in its past 33 games, has lost three of five.
Atlanta held leads of 1-0 on Wheeler's 15th goal in the second and 2-1 on Byfuglien's 20th goal in the third, but Kovalchuk tied it with 12:09 remaining.
New Jersey is 11th in the Eastern Conference, a point behind the Thrashers.
Wheeler beat Brodeur to the glove side 10:14 into the second with a wrister from the right circle. Andrew Ladd assisted with a pass across the slot.
The next four goals were scored on power plays.
Zajac's 11th goal, which made it 1-1, came 2:13 later and 1:35 after New Jersey began a two-man advantage courtesy of high-sticking penalties on Atlanta defensemen Tobias Enstrom and Chris Thorburn.
Thrashers goalie Ondrej Pavelec stopped three shots in New Jersey's 5-on-3 before Patrik Elias passed from the bottom of the right circle and Zajac scored from the left side of the crease.
"It just came down to special teams," Atlanta coach Craig Ramsay said, "and we didn't get the job done."
Byfuglien's slap shot from the right side of the blue line beat Brodeur's glove side. The New Jersey bench wanted goalie interference called on Nik Antropov, but it appeared Devils defenseman Colin White knocked down the Atlanta center while Byfuglien fired the shot.
"He was laying on me when [Byfuglien] shot the puck," Brodeur said. "He hit my glove, yanked my leg and the net. He's a guy that does that, but it's easier to digest when you win the game."
Kovalchuk, after lifting the puck under the crossbar, celebrated his 10th goal in his past 15 games by slamming into the rear boards with a big smile as Atlanta fans booed their one-time franchise star and No. 1 overall draft pick of 2001.
"It doesn't bother me," Kovalchuk said. "Everything I do, they boo me."
The former Thrashers captain, who was traded to the Devils last season and last summer signed the NHL's richest contract, had an uncontested look at the net after controlling the rebound of Mark Fayne's slap shot.
"It was a big goal for us," Kovalchuk said. "It tied the game. It was nice. It was an easy one."
On the winning goal, Zajac scored with a wrister beneath the left circle. Kovalchuk began the play and earned an assist with a pass to Elias, who controlled the puck with his skate in the right circle before flicking it across the slot to Zajac.
"For us, we stuck with it," Zajac said. "We didn't give them too many shots. It was a tight game. Our power play came through for us."
Brodeur made his sixth straight start, improving to 5-1 in his past six and 18-20-2 overall. He stopped 24 of 26 shots for his NHL-leading 620th career win.
Pavelec dropped to 19-19-8. He made 27 saves.
"You have to play smarter," Pavelec said. "We did not play smart enough to win the game."
The Devils have won three straight and 11 of 14 on the road, and have played eight consecutive one-goal games.
"We came back, we battled and just kept playing, coming from behind twice," New Jersey coach Jacques Lemaire said. "We needed this game."
Evander Kane's tripping penalty gave the Devils a man-advantage in overtime. "I took a late penalty," he said. "That's on me, and they scored. Obviously, I didn't agree with it. It cost us the game." ... The Devils held an opponent scoreless in the first period to extend their modern-day NHL record to 15 straight games. They have played eight consecutive 0-0 first periods.
Copyright by STATS LLC and The Associated Press
Arena: Philips Arena Location: Atlanta, Georgia
Referees: Brad Meier, Dan O'Rourke Linesmen: Steve Barton, Jean Morin
Attendance: 16,073 (86.7% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
No scoring this period
Blake Wheeler (15)
Assists: Andrew Ladd, Bryan Little
Travis Zajac (11)
(Power Play) Assists: Patrik Elias, Brian Rolston
Dustin Byfuglien (20)
(Power Play) Assists: Tobias Enstrom, Bryan Little
Ilya Kovalchuk (25)
(Power Play) Assists: Nick Palmieri, Mark Fayne