Tampa Bay got goals from Evgeny Artyukin, Steven Stamkos, Ryan Malone and Martins Karsums.
Brodeur was scheduled to have a quiet night after being honored in a pregame ceremony for setting the NHL record for regular-season wins. That changed when backup Kevin Weekes injured a leg in the opening minute of the second period.
On the winner goal, Parise took a cross-ice feed from Travis Zajac and drilled a shot into the net before goalie Karri Ramo could react at 1:21.
Brodeur unexpectedly played a role in sending the Lightning to their fifth straight loss.
Artyukin, who gave the Lightning the lead on the first shot 1:38 into the game, fired a puck from the left wing that appeared to hit Weekes on the left leg above the goal pad. Weekes made the stop but pitched forward, facedown on the ice. He was eventually helped off, unable to put weight on the leg.
Brodeur came on and surrendered a power-play goal to Stamkos at 5:09, giving Tampa Bay a 2-0 lead. The tally was Stamkos' 21st, equaling the club record for a rookie held by Brad Richards.
Shanahan's goal tilted the momentum in New Jersey's favor. The penalty shot was awarded when Ramo fell while attempting to clear the puck behind his net. As the puck rolled toward the wide-open net, Ramo threw this stick, attempting a deflection. He missed, but Shanahan didn't on the subsequent penalty shot - drilling a blistering shot between Ramo's pads at 13:05.
Pandolfo followed with the tying goal at 14:50, a drive from the left wing over Ramo's left shoulder. Langenbrunner added a power-play goal with 1:31 remaining to give the Devils a 3-2 lead.
The Lightning struck first in the third, making it 3-3 as Malone broke in alone to beat Brodeur with a backhander at 2:41.
Langenbrunner countered for New Jersey with his second goal, knocking Zajac's rebound past Ramo at 11:35.
Karsums tied it 4-4 at 14:06; pouncing on a rebound in the slot.
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