John Madden scored two goals for the Devils, who chased former Michigan teammate Marty Turco from the net in the second period.
Zach Parise and David Clarkson had goals 53 seconds apart in the third period, and Jamie Langenbrunner gave New Jersey a 1-0 lead in the Devils' best offensive output this season.
After a one-game commitment to defense against the New York Rangers in a 2-1 victory Wednesday, the Stars fell back into bad habits. Turco left after allowing three goals on 17 shots in 30:17 of action, and Tobias Stephan gave up a pair of quick ones in the third.
Dallas (2-4-1) has yielded 29 goals - more than four per game.
Brodeur is five shutouts away from Terry Sawchuk's NHL record of 103 and eight wins shy of Patrick Roy's league mark of 551.
Langenbrunner made it 1-0 just 1:59 in, and Madden scored a pair 1:12 apart in the second. Turco was replaced by Stephan after Madden's second at 10:17 Avery made headlines last spring during the playoffs when he tried the unorthodox approach of waving his arms and stick in the face of Brodeur while he and the Rangers were on a power play.
A rule eliminated the practice the following day, but couldn't stop the battle between the Vezina Trophy winner and the NHL's most despised player. Brodeur refused to shake Avery's hand after the Rangers knocked New Jersey out in the first round.
Brodeur gave Avery a few shoves to try to move him out of the crease in their latest meeting, and Avery complained that penalties weren't called. Brodeur also fired a puck at Avery during a clearing attempt from behind the net in the second period.
Avery also renewed hostilities with Clarkson, who was called for interfering with him in the first period. Avery and Clarkson had a pregame skirmish last November when Clarkson stepped in on Avery, who was jawing with Brodeur.
Clarkson got some shots in on Avery after exiting the penalty box and seemed ready to drop the gloves, but Avery skated to the bench. The two kept bickering there.
Avery also stopped to talk to several fans in Devils jerseys, who yelled down to him as he walked into the tunnel toward the dressing room after warmups.
Avery has had a busy week. He made his return to Madison Square Garden on Monday and heard mostly boos during Dallas' win. The booing, hissing and derogatory chants and taunts followed him across the river to New Jersey.
He will conclude his tour of the metropolitan area Thursday when the Stars face the New York Islanders.
Brodeur already has two shutouts in six games this season. This time, he was the beneficiary of a boatload of goals - at least by New Jersey standards. The Devils are 5-1 despite scoring only 15 goals. After scoring six in the first four games, New Jersey has nine the past two.
His first big save came against Avery, who quickly one-timed a shot in the slot in the first period. Brodeur also flopped on his back into the net, while holding his glove behind his pads to keep the puck out in the second, and denied Mike Ribeiro on a breakaway in the third.
Notes: Brodeur beat Atlanta 1-0 last Thursday. ... Devils enforcer Pierre-Luc Leblond earned his first assist in his first NHL game. ... Stars D Doug Janik (facial cuts) missed his second straight game.