Zach Parise and John Madden each had a goal and an assist for New Jersey, which scored two goals short-handed. The Thrashers were 0-for-7 on the power play, leaving them 1-for-32 in the last nine games.
"When you go minus-2 on your power play, it's not a good sign," Thrashers coach John Anderson said.
Anderson said "it could have been a different game" if his team had "been sharper" on the power play, but the Devils threatened to turn the game into a rout in the first period.
Travis Zajac scored 4:34 into the game, Madden scored short-handed on a shot over Kari Lehtonen's glove and David Clarkson scored in front of the net for a quick 3-0 lead 11:42 into the game.
Colby Armstrong scored Atlanta's only goal early in the second period, but the Devils shut down any hopes of a comeback by answering with a power-play goal by Parise and Brian Gionta's unassisted goal, also short-handed.
"Those (short-handed) goals were great," Clarkson said. "The guys on the power play did a great job. Our special teams worked hard."
The Devils won three nights after a loss to Washington snapped their winning streak at eight games. The Thrashers have lost five of six.
Though New Jersey leads the Atlantic Division and the Thrashers are last in the Southeast Division with 41 points, second-lowest in the NHL, the Devils felt they had a point to prove.
Atlanta's only shutout win of the season was a 4-0 victory by Lehtonen at New Jersey on Jan. 8. Brent Sutter said the loss may have been the worst in his time as New Jersey's coach.
"This was big," Clarkson said. "These guys beat us the last time we played them.
"We gave everything we had right away."
Sutter said his team "did a good job," though there was no scoring after the second period.
"We sat on the lead a little bit in the third period," Sutter said.
The Thrashers had an 18-3 advantage in shots on goal in the third period, to finish with a 32-31 edge, but didn't score.
Sutter, who pulled goalie Scott Clemmensen in the second period of last month's loss to the Thrashers, gave Weekes his first start since Jan. 16.
"I try not to think about that," Weekes said of the long wait between starts. "I don't pay much attention to how many days go by. I go out hungry and do my best."
New Jersey's defense gave Weekes ample time to become comfortable in his return to the ice. The Thrashers managed only four shots on goal in the opening period. The Devils' 3-0 lead came on 18 shots on goal.
"We were able to pen them up," Weekes said.
"Our guys blocked a lot of shots. We did well on defense in the first two periods. ... This win is reflective of the character of this team."
Lehtonen stopped 19 shots, but he was pulled after giving up five goals in 28 minutes.
"I don't think he was very sharp tonight," said Anderson of Lehtonen.
"He didn't stand a chance on a couple of those goals. We need him to make big saves for us. We are that type of hockey team."
Johan Hedberg replaced Lehtonen in the net for Atlanta and stopped seven shots.
Notes: Parise's power-play goal in the second period ended a streak of 18 penalty kills over six games by Atlanta, including the Devils' first two chances. ... Atlanta's Eric Boulton traded punches with New Jersey's Brendan Shanahan late in the first period. ... The Devils have won seven straight road games and have four straight wins in Atlanta.