"It always comes back to bite you when you play that poorly and take a penalty," Stars coach Lindy Ruff said.
New Jersey allowed only 16 shots and scored twice on the power play.
"Huge, two power-play goals," New Jersey coach Peter DeBoer said. "It probably didn't need to get to overtime if we could have buried some of our chances, but we did on the power play and that was critical."
The Stars tied the score on Tyler Seguin's goal with 52 seconds remaining in regulation play.
Jamie Benn scored on a backhand at 10:52 of the second to cut the deficit to 2-1.
The Devils, allowing the fewest shots per game of any NHL team, held Dallas to its season low. Cory Schneider made 14 saves for his second victory over the Stars this season.
Elias and Zajac each had an assist and Jaromir Jagr assisted on two goals for New Jersey.
Seguin added an assist for Dallas. Kari Lehtonen made 21 saves. He has allowed only six goals in losing three of his last four games against New Jersey
The Devils dominated early. Dallas didn't put a shot on goal until a 5-on-3 advantage more than five minutes in. Even during that 75-second stretch, the Stars had only one shot.
"I'm embarrassed myself with that," Seguin said. "That's unacceptable."
Ruff said: "It was just a killer."
DeBoer called the 5-on-3 penalty kill "huge."
"It would have been easy there to lose our focus. We didn't. We buckled down. To get out of the first period up was key."
New Jersey scored first, at 10:59 of the first period. Zubrus sent a slap shot from the right faceoff circle over Lehtonen's glove and into the right corner.
Dallas had its best scoring chance a minute and a half before the end of the first period. Cody Eakin took a pass in the slot but shot the puck into Schneider's body.
For the period, New Jersey outshot the Stars 7-4. The four shots tied Dallas' season low for a period.
Lehtonen prevented another Devils goal three minutes into the second period when he made a glove save on Zajac while prone on the ice.
Zajac did score on a power play at 9:50 of the second, when he tipped Damien Brunner's drive past Lehtonen.
Barely a minute later, Dallas broke through for its first goal against Schneider in more than 90 minutes this season. Benn took a pass from Seguin with his back to the goal and backhanded the puck into the net.
New Jersey again had an advantage in shots, 7-6, with the Devils' final two shots short-handed
Schneider has the only shutout against Dallas this season, with 26 saves in a 1-0 victory Jan. 9.. The Devils' season low in shots allowed for a game is 15 ... In the Stars' previous seven games they averaged 33 shots on goal ... Dallas' fewest shots in a game this season had been 18 in a 4-3 victory at Chicago Dec. 3. ... New Jersey won both games between the teams this season ... The Stars' streak of scoring a power-play goal in eight consecutive games ended ... Play was stopped during the third period because a panel of glass broke above the boards ... Dallas recalled C Dustin Jeffrey from Texas of the American Hockey League Thursday, but he was inactive for the game.