NEWARK, N.J. -- Down a goal late in the third period against one of the better teams in the National Hockey League, the New Jersey Devils refused to flinch.

Instead, the surprising Devils received a clutch, game-tying goal from Drew Stafford and then a game-winning tally by Kyle Palmieri in the shootout to rally past the Tampa Bay Lightning 5-4 on Tuesday night at the Prudential Center.

"It's a learning process to win these types of games," said Palmieri, who scored the only goal in the three-round shootout. "If you look back at last year, it was a disappointing year, but we turned the page and are off to a good start and hopefully can continue to play this good hockey."

Stafford scored twice for the Devils (5-1-0), who killed off a penalty in the final two minutes of regulation and another with under a minute to play in overtime.

Goaltender Cory Schneider made 33 saves for the Devils, stopping all 15 shots he faced in the third period and overtime.

"It was a couple huge kills, the penalty kill stepped up great," Palmieri said. "Sometimes your best penalty killer has to be your goalie, and (Schneider) without a doubt did that. He played a great game."

Steven Stamkos had a season-high three points with a goal and two assists for the Lightning (5-1-1), who had a four-game winning streak snapped. Tampa Bay's Nikita Kucherov scored his eighth goal and added an assist. Both players have earned at least one point in all seven games this season.

The teams combined for five goals in a wide-open first period that saw the Devils emerge with a 3-2 lead.

Lightning goaltender Peter Budaj (30 saves) allowed a soft goal to Stafford just 2:45 into the game, but the visitors answered with quick goals from Vladislav Namestnikov -- his third of the season -- at 7:33 and Ondrej Palat -- also his third -- at 12:21.

Palmieri ripped a right-wing slap shot past Budaj to tie the score at 16:19, his second goal of the season. Brian Gibbons then restored the Devils' one-goal lead with 51.5 seconds left before the first intermission, potting a backhand shot for his third goal.

Budaj settled down in the second period, and the Lightning grabbed a 4-3 lead with Kucherov and Stamkos finding the back of the net.

Kucherov tied the score by blistering a left-wing slap shot past Schneider at 14:56 of the second period. Kucherov has at least one goal in all seven games this season, five shy of Mario Lemieux's NHL record to start a season.

"He's an impressive player," Lightning right winger Ryan Callahan said of Kucherov. "Every year I've been here, it seems like he's getting better and better. This year, he's stepped up to another level."

Less than three minutes after Kucherov's goal, Stamkos was left alone in the slot. He picked up a loose puck that squirted out front from behind the Devils net and buried a backhand for his second goal of the season at 17:23 to make it 4-3 Lightning.

Schneider made a few terrific saves to keep the Devils within one before Stafford converted a power-play rebound at 15:54 of the third period, his second of the night and third goal in the past two games.

"For the most part, goals are scored in those greasy areas, and it's a matter of us funneling pucks on net and grinding out those bounces," Stafford explained. "Sometimes you get a lucky break, it bounces off a skate or a rebound bounces open, and if you have bodies in front, good things happen."

Despite the defeat, the Lightning earned three out of four points playing on consecutive nights, coming off a 3-2 road win over the Detroit Red Wings on Monday.

"Ultimately, I look at the big picture and say we got three of four points and I'm pretty happy with the way things went," Lightning coach Jon Cooper said. "This team gives an honest effort. Parts of the game we can tighten up, there's no question. The guys are working on it."

NOTES: LW Ondrej Palat scored a power play goal in the first period, giving the Lightning at least one power-play goal in all seven games this season. ... The Lightning scratched RW J.T. Brown, D Slater Koekkoek and C Gabriel Dumont. ... The Devils scratched D Ben Lovejoy, D Dalton Prout and RW Stefan Noesen.