Patrik Elias registered a goal and an assist for the Devils, who won for just the second time in their last six games. David Clarkson and Jay Pandolfo also scored, while Dainius Zubrus had two assists.
With franchise goaltender Martin Brodeur sidelined for an elongated period, veteran Kevin Weekes will be counted on to handle the starting duties in the four-time Vezina Trophy winner's absence, unless Devils general manager Lou Lamoriello makes a move to acquire a proven netminder.
A second-round (41st overall) pick by Florida in 1993, Weekes made 24 saves to pick up his first win since November 12, 2007.
Brodeur is scheduled to undergo surgery on Thursday to repair the distal biceps in his left elbow and will be out 3-4 months. He suffered the injury in Saturday's 6-1 triumph over Atlanta.
One of the most celebrated goaltenders in league history, Brodeur has collected 544 career wins, seven shy of Patrick Roy's NHL record, and needs five shutouts to equal Terry Sawchuk's record of 103.
Jussi Jokinen lit the lamp twice for Tampa Bay, which had a three-game winning streak snapped. The Lightning, though, have still recorded a point in eight of their past nine games, going 4-1-4.
Mike Smith, among the league leaders in save percentage, stopped 35-of-38 shots in defeat.
Tampa Bay tied the game at two on the power play, as Jokinen one-timed a cross-ice pass from Andrej Meszaros just inside the far post with 9:08 remaining in the third period.
The Devils regained the lead on Pandolfo's tally with 4:50 to play in regulation. Elias let fly a slap shot from inside the right circle. Smith made the save and tried to cover the puck, but John Madden poked it loose and Pandolfo shoveled the disc past the sprawled netminder.
New Jersey defenseman Sheldon Brookbank was assessed a delay of game penalty for firing the puck over the glass 18 seconds later and the Lightning capitalized on the ensuing power play. Tampa Bay captain Vincent Lecavalier's blast from the right point was kicked out into the middle by Weekes and Martin St. Louis banged in the rebound with 3:20 left on the clock.
Both teams generated four shots in overtime.
Lecavalier beat Weekes on the glove side to begin the shootout, then New Jersey's Zach Parise fired a shot that clanged off the cross bar. After Jokinen was denied, Elias faked the forehand to freeze Smith before scoring on the backhand. Weekes made a right pad stop on Vaclav Prospal and Langenbrunner beat Smith on a backhand-to-forehand deke to earn the extra point.
Jokinen gave the Lightning a 1-0 lead with 3:32 left in the opening period. The towering Evgeny Artyukhin lumbered down the right wing and threw a backhander on goal that Weekes stopped. Mark Recchi gathered the rebound at the right side of the net before dropping the puck to Jokinen, who was alone in the slot and able to tuck it home.
New Jersey tied the game with 3:55 to go in the middle stanza, as Clarkson made his way down the right wall and fired a shot from a sharp angle that beat Smith high on the short side.
Elias put the Devils up 2-1, scoring a power-play goal over Smith's left shoulder with 1:16 remaining.