The Devils ended a two-game losing streak to the Stars with their ninth win in 11 games.
Dallas has won only one of its last nine away from home (1-5-3).
New Jersey opened the scoring when Elias tipped Brian Rolston's point shot past Alex Auld with 1:58 remaining in the first period.
In the second, Dallas had excellent opportunities to tie it after Elias was whistled for cross-checking at 4:43, but Brodeur denied Loui Eriksson, and Trevor Daley hit the post.
"At that time it was a big save," Brodeur said of the stop on Eriksson. "It could have been different from there. We finished off the penalty kill and we took over the game."
The failed power play was costly as Elias, just out of the penalty box, fired a shot over Auld's right shoulder to give the Devils a 2-0 lead at 6:59.
"A goal there to tie the game would have been huge." Auld said. "Then they come back and score. I don't know if you call it a turning point, but you want to build momentum of the power play. Obviously, the timing is tough because they seemed to gain a lot of momentum from that."
Zajac then ended a 12-game goal drought with a power-play tally at 9:36. Stars coach Marc Crawford called his timeout in an effort to settle the team.
It didn't help, as Parise expanded the Devils lead to 4-0 at 18:11.
The focus then turned to preserving the shutout. The Stars generated little sustained pressure in the final period as Brodeur made a sharp-angle stop on Tom Wandell and a pad save on Eriksson.
Elias took a tripping penalty with 3:34 left, giving Dallas one final shot to score. Brodeur was up to the challenge, stopping Brad Richards and Mike Modano.
"I don't think we had a whole lot of chances in the game and we lost some puck battles that ended up giving them the chances that they needed," Crawford said. "I thought we were good in the first and salvaged a little bit of pride in the third. The turnovers and the lost puck battles were the difference."
The Devils placed RW David Clarkson on injured reserve and activated LW Pierre-Luc Letourneau-Leblond from that list. Clarkson played two games after missing 13 because of a non-displaced fracture above the right ankle. He will miss another two to three weeks after aggravating the injured area, according to general manager Lou Lamoriello. "Let's just make sure everything is healed up without any question. No chances will be taken," Lamoriello said. ... Lamoriello said C Dainius Zubrus, sidelined 21 games because of a fractured right kneecap, will resume skating soon but is still "a few weeks away." ... The Devils honored their 2000 Stanley Cup winning team that wrested the title away from Dallas. Only Brodeur, D Colin White, LW Jay Pandolfo and Elias remain from the squad that beat the Stars in six games.
Copyright by STATS LLC and The Associated Press
Arena: Prudential Center Location: Newark, New Jersey
Referees: Kerry Fraser, Stephane Auger Linesmen: Michel Cormier, Steve Miller
Attendance: 14,202 (80.6% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Patrik Elias (9)
Assists: Brian Rolston, Dean McAmmond