New Jersey's Kyle Palmieri was the lone shootout taker to convert, slipping a shot past Lightning goalie Peter Budaj in Round 3 to secure the win for the Devils.
Brayden Point, Nikita Kucherov and Ryan Callahan all missed on their shootout attempts.
The Lightning rallied from a 3-2 deficit after the first period and led 4-3 into the second intermission and through much of the third. Drew Stafford's late power-play goal with 4:06 to go in regulation proved to be the difference between the Bolts getting one point or both.
Through two games of a three-game road trip, the Lightning have gathered three of a possible four points.
"Can you look at it now and say, 'Hey, it would have been great to get two (points)?'" Lightning head coach Jon Cooper said after the game. "Yeah, for sure, but we took three of four and now it just puts the onus on us to get some points in Columbus."
Tampa Bay closes out its road trip with another tough foe as the Blue Jackets are tied with New Jersey atop the Metropolitan Division.
Columbus won all three meetings against the Lightning last season.
But before looking ahead to that big matchup, we'll review Three Things that got the Lightning a point in New Jersey but kept them from grabbing two.
Video: TBL@NJD: Kucherov extends goal streak, ties game1. KUCHEROV CONTINUES TORRID SCORING PACE
Nikita Kucherov has already established a new Tampa Bay franchise record for consecutive games with a goal to start a season.
That came in game four against St. Louis.
Now his goal-scoring streak is at seven games and shows no signs of stopping.
Kucherov scored his eighth goal of the season at 14:02 of the second period to tie the game 3-3. And just like most of his goals this season, No. 8 was a thing of beauty. Steven Stamkos carried the puck into the zone, drawing defenders his direction. He dished off to his left for an open Kucherov, who wound up at the top of the left circle and lasered a shot off the post and in, Cory Schneider, one of the NHL's top if not overlooked goalies, too slow to react.
Kucherov is in elite company with his marker on Tuesday, becoming one of just two players in the NHL over the last 30 years to score in seven-consecutive games to start a season, joining Mario Lemieux, who scored in 12 straight to open 2002-03.
Kucherov has work to do to match Lemieux's incredible start.
But at this point, can you really doubt his ability to get there?
Every time Kucherov touches the puck, there's a realistic expectation it could end up in the back of the net.
He's the hottest player in the NHL currently, and if you had to point to just one reason why the Lightning are off to such a great start, for me, it's the play of Nikita Kucherov.
Video: TBL@NJD: Budaj robs Hischier's excellent chance2. BUDAJ BATTLES BACK
The first period Tuesday night is one Bolts backup goalie Peter Budaj would probably like to forget.
In his first action of the season after watching Andrei Vasilevskiy play every minute through the first six games, Budaj had considerable rust to shake off. On just the second puck he faced, Budaj allowed an innocuous-looking shot to go through his legs. Later, Kyle Palmieri lifted a slap shot from the edge of the right circle over Budaj's glove as the Lightning goalie was late to react.
In all, Budaj gave up three goals in the opening period, and the Lightning trailed 3-2 after 20 minutes.
"It's always tough on a guy when you, it's the gamble you take when you don't play a guy for a while," Cooper said. "But the big thing for me is how he responded. I'm sure he wants to have that first one back, but he made some big saves for us when we needed them."
Budaj was a much more confident goalie from the second period on. The Bolts high-octane offense certainly helped him settle in too.
Nikita Kucherov and Steven Stamkos netted back-to-back goals to put the Lightning back in front in the second period.
And then Budaj went to work keeping them there. With 2017 No. 1 overall draft pick Nico Hirschier searching for his first goal, Budaj made a fantastic sprawling save to tip Hirschier's Grade-A scoring chance over the crossbar and preserve the lead.
In the third, Budaj denied Drew Stafford on a 3-on-2 break and then stonewalled a wide-open Hirschier at the edge of the crease seconds later.
A late power-play goal by Stafford -- a tally aided by a hurting Anton Stralman, who blocked a shot seconds earlier and couldn't keep up with Stafford on a gimpy wheel - prevented Budaj from seeing out the victory.
But the way he rebounded from a less-than-stellar opening period should give the Lightning plenty of confidence they can turn to their backup when necessary and not see much, if any, dropoff from starter Andrei Vasilevskiy.
Video: Cooper | Postgame NJD 5, TBL 43. SPECIAL TEAMS FALTER
Yes, Tampa Bay saw its streak of power-play goals to start the season improve to seven games, tying a franchise record set in 2000-01 and 2002-03, after Ondrej Palat's man-advantage marker in the first period.
But the Lightning had a couple opportunities late to break a 4-4 tie on the power play and couldn't convert.
And the Bolts gave up a power-play score trying to preserve their one-goal lead with 4:06 to go to miss out on both points.
The Lightning were handed a power play for nearly the final two minutes of regulation when Pavel Zacha took exception to Jake Dotchin laying out Brian Gibbons in front of the Bolts' net and went after the defenseman, drawing a roughing penalty.
The Lightning were unable to generate much of anything with the late gift, however, and the game headed to overtime.
In the extra session, the Bolts benefitted from another late whistle, getting a 4-on-3 advantage for the final 48 seconds of overtime.
Again, the Lightning couldn't take advantage.
Meanwhile, New Jersey scored on half of its power-play opportunities, netting two goals on four chances, including Stafford's game-tying goal with four seconds remaining on a man-advantage with time running out in the third period.
The Lightning have generally been pretty good on special teams this season, and their power play has been exception to start.
But New Jersey won the special teams battle on Tuesday, and the Bolts lost out on a point as a result.