Tampa Bay (49-24-8) trails the Montreal Canadiens by two points for first place in the Atlantic Division. Each team has one game remaining; the Lightning hold the tiebreaker with 47 regulation and overtime wins compared to the Canadiens' 43.
Johnson scored the overtime winner on a 4-on-3 power play when his shot was deflected by Devils defenseman Seth Helgeson and got past goalie Keith Kinkaid. It was Johnson's second goal of the game, 29th of the season.
"I was trying to get the puck to [Palat] and I tried to get it between the defenseman's legs, but it deflected off his leg and went in," Johnson said. "Sometimes you get lucky. I like to say it's because of geometry. I could say I tried to do it, but maybe it was just the right luck at the right time."
The Lightning scored three power-play goals.
"You need that power play going," Tampa Bay coach Jon Cooper said. "We had eight of them and one was a 4-on-3. They were obviously big goals for us; the last one is a winner. You need to have your special teams going."
"Being on the penalty kill in overtime is tougher because it goes on to 4-on-3," Helgeson said. "It was just a tough play at the end."
Bernier gave the Devils a 1-0 lead at 5:06 of the first period when he took the puck off a face off and sent a wrist shot past Lightning goalie Ben Bishop on the stick side.
The Lightning rallied to take a 2-1 lead by the end of the first period.
Johnson scored at 6:26 when Kucherov skated to the left circle and faked a shot that took Kinkaid out of position. Kucherov then slid a pass to Johnson, who was standing in front of the open net for the easy tap-in goal.
"Playing with [Kucherov] and [Palat], you have to expect they are going to be getting the puck and you have to be ready when they do because they are going to make something happen," Johnson said. "They are a lot of fun to play with."
Nikita Nesterov gave the Lightning the lead with a power-play goal at 9:13. Nesterov took a pass from Johnson at the top of the point and sent a slap shot past Kinkaid with Ryan Callahan providing a screen.
"We had too many penalties," Devils interim co-coach Lou Lamoriello said. "You can't take eight penalties, you know, especially against a team like that. But our effort was excellent."
Elias tied the game 2-2 at 10:17 of the second when he scored off a rebound in the slot after Bishop made a save on Stephen Gionta's initial shot.
Lightning captain Steven Stamkos scored his 43rd goal on another power play at 17:42 of the second with a one-timer from the left circle that beat Kinkaid on the glove side to put Tampa Bay ahead 3-2.
"We took awhile to get things going, but we've got to be better," Stamkos said. "That's a team that's really not playing for much right now, and we had a couple of lapses in the defensive zone, and that's something we don't want to do come playoff time.
"But it was nice to see the special teams come through on a night that we didn't play too well. We've probably lost some points this [season] because the power play wasn't going."
Bishop made 21 saves, and Kinkaid 27.
Tampa Bay will close out the regular season at home against the Boston Bruins on Saturday. The Bruins lost 4-2 to the Florida Panthers on Thursday and trail the Pittsburgh Penguins by one point for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"There's plenty on the line for both teams, maybe a little more for them," Stamkos said. "But we have a chance to knock them out and challenge for the division title, which would give us home ice at least until the conference finals."