Mathieu Garon stopped both the New Jersey attempts in the shootout to give Jon Cooper a win in his debut as Tampa Bay coach.
The Lightning twice overcame a two-goal deficit to force overtime with Steven Stamkos scoring twice and running his season total to 25.
Ryan Carter scored shorthanded for the Devils 2:39 into the third period to give his team a 4-2 lead, but Stamkos brought the Lightning within one with 12:49 to play and Alex Killorn tied it with just 16 seconds to go in regulation when he put a slap shot past Martin Brodeur.
Brodeur then allowed the two quick goals in the shootout while Garon was perfect in the net. Brodeur took the loss while making 21 saves. Garon also stopped 21 shots.
Cooper took over the Lightning after Guy Boucher was fired earlier this week. Cooper had been coaching the Tampa Bay minor league team in Syracuse.
Dennis Rodman pledges to end trips to North Korea
Susan Sarandon 'very excited' about daughter's pregnancy