Goaltender Keith Kinkaid poke-checked the puck off Dylan Larkin's stick 1:43 into overtime after he got behind the Devils defense, and Gibbons scored 49 seconds later after a turnover by Red Wings defenseman Nick Jensen.
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"I thought we played an intense game, and I'm happy with the way we finished it," coach John Hynes said. "We made some mistakes in the third, and Keith had to make some big saves right after they tied it, but we settled down and won the game."
Brian Boyle, Taylor Hall and Nico Hischier scored for New Jersey (14-5-4), and Kinkaid made 31 saves.
"This is a pretty resilient team," Kinkaid said.
The Devils won for the third time in four games (3-0-1), including a 3-2 victory against the Vancouver Canucks on Friday.
"This was a very evenly matched game, and we both played last night, so we had to step up at the end," Boyle said.
David Booth scored two goals for the Red Wings (10-9-5), who have lost four in a row (0-1-3), and Anthony Mantha had a goal.
Video: NJD@DET: Booth gathers loose puck, scores his second
Petr Mrazek made 21 saves on 24 shots before leaving 6:12 into the third period because of a possible concussion. Jimmy Howard made seven saves before allowing the overtime goal.
"It was a tight game, and I liked the way we battled back," Red Wings captain Henrik Zetterberg said. "It's just tough to lose another overtime game."
Booth, playing for the first time since Nov. 15, gave the Red Wings a 1-0 lead 34 seconds into the game. Xavier Ouellet took a shot with Luke Glendening and Booth in front of the net. Glendening tipped it an instant before Booth redirected it past Kinkaid.
It was Booth's first NHL goal since he scored for the Toronto Maple Leafs on March 31, 2015, against the Tampa Bay Lightning. He played the past two seasons in the Kontinental Hockey League before signing as a free agent with Detroit on Oct. 2.
"It's a great story," Red Wings coach Jeff Blashill said. "David was a 30-goal scorer in this league, and then he was out of the league and almost out of hockey before he made our team in camp. He's worked hard for this."
Boyle tied it 1-1 at 9:12, tipping Steven Santini's shot past a stranded Mrazek for his second goal in as many nights.
Video: NJD@DET: Boyle scores on terrific deflection
Hall gave the Devils a 2-1 lead with a power-play goal at 13:21 when he scored on a wrist shot from the slot.
New Jersey forward Travis Zajac put a rebound into an open net at 3:56 of the third period, but Adam Henrique was called for goaltender interference, negating the goal.
The Devils killed the ensuing power play, and Hischier made it 3-1 at 6:12, beating Mrazek after a Larkin turnover.
Video: NJD@DET: Hischier finishes great Hall's great pass
"[Henrique] made a great play to keep the puck below the goal line, and Hall gave me a great pass," Hischier said. "I just had to finish."
Mrazek, who was shaken up in the collision with Henrique, was replaced by Jimmy Howard after the goal under the League's concussion protocol.
"I know the rule, but I didn't have any pain in my head," he said. "I just lost my breath for a little bit."
Mrazek stayed in the game for 2:16 between the collision and Hischier's goal.
Mantha made it 3-2 at 6:50, beating Kinkaid to end a four-game point drought, and Booth tied it 3-3 with his second of the game at 8:16.
Goal of the game
Booth's goal 34 seconds into the first period, when he and Glendening tipped Ouellet's shot from the right point.
Video: NJD@DET: Booth deflects early shot past Kinkaid
Save of the game
Howard smothered Henrique's one-touch redirection from the right hash marks off a pass from Jesper Bratt with 7:37 remaining in the third period and the Devils on the power play.
Video: NJD@DET: Howard smothers deflection by Henrique
Highlight of the game
Mrazek's save on Bratt at 7:09 of the first period.
Video: NJD@DET: Mrazek uses his pads to usher Bratt aside
They said it
"I didn't think the goal was going to count, but the guys told me to throw my arms in the air and start celebrating, so that's what I did." -- Devils forward Travis Zajac on his disallowed goal in the third period
"This was a special night. I've had a long career, with some ups and downs, and I think a lot of people had given up on me. At times, I had given up on myself." -- Red Wings forward David Booth on scoring his first NHL goals since March 31, 2015
Need to know
Kinkaid allowed a goal in the first 41 seconds each of his past two starts but won both. … The Red Wings are 0-4 in games decided in overtime. The Devils are 3-2.
Devils: Host the Florida Panthers on Monday (7 p.m. ET; MSG+, FS-F, NHL.TV)
Red Wings: Host the Los Angeles Kings on Tuesday (7:30 p.m. ET; SNE, SNO, FS-D, FS-W, NHL.TV)
Video: Hall, Gibbons lead Devils to 4-3 win in overtime