Brodeur earned his third shutout of the season and 124th of his career.
Jaromir Jagr assisted on both goals to raise his assist total to 1,024.
Damien Brunner's seventh goal had broken the scoreless tie in the second period.
Jagr skated behind the net and passed to Zubrus on doorstep of the crease for the goal at 6:44 of the period.
Then at 9:36, Jagr put a backhand shot on goal and Zubrus was there to knock in the rebound for his seventh goal and a 3-0 advantage.
Brodeur came up with two of his best saves in the third to thwart the Lightning.
Brunner scored at the 15:37 mark. Rookie defenceman Eric Gelinas made a pass up the middle to Brunner's stick, just before the blue line.
The Lightning put pressure on Brodeur in the second period, outshooting the Devils 14-5.
Before Brunner's goal, Brodeur had Martin St. Louis muttering to himself in the second period after the Devils goalie foiled a scoring chance by the Lightning's star.
St. Louis received a pass in front and wristed a high shot from the point that Brodeur managed to grab out of the air with his glove.
In the first minute of the third period, Brunner was denied a second goal when Lindback came up with a big save on a shot from the right circle.
NOTES: Devils coach Pete DeBoer named Marty Brodeur as his starter vs. the Lightning and then added that he's looking for one of his goalies to get hot. "I'm not married to an even split," DeBoer said. "If one of them gets on a roll, I'm going to run with them. Both of them have played well and lost. Winning games, that's what it comes down to." ... Tampa Bay star Steve Stamkos joined his teammates for the morning skate at the Prudential Center on Saturday, 33 days after breaking his leg in a collision with a goal post. "My 4-week X-ray looks good. I'm just trying to progress," said Stamkos in a video on the Lightning website. Stamkos had 14 goals and 23 points in 17 games before the Nov. 11 injury sustained in Boston.