Travis Zajac, Miles Wood, Kyle Palmieri and Brett Seney also scored for the Devils, who trailed 3-0 after 9:23 and 4-1 at 3:57 of the second period.
"It wasn't a dream start by any means, being down three," said Palmieri, who scored his New Jersey-leading 17th goal to make it 4-3 at 6:24 of the third period.
"It was one of those things where we didn't really know what [Vegas] could handle as a team over there. Once we got to our game, we knew we could control the play and get scoring chances. We had a ton of scoring chances in the first and finally broke through in the second, and I think throughout our lineup we had contributions.
"We were missing a pretty important part of our team and you are going to need that from everybody. Everybody contributed tonight. It's a fun win and we are going to enjoy it."
The Devils played without forward Taylor Hall (lower body), who leads New Jersey in scoring with 31 points (eight goals, 23 assists) in 29 games.
William Karlsson, with two goals, Alex Tuch and William Carrier scored for Vegas (18-14-2), which is 9-2-1 in its past 12 games.
"After we scored that third one, we just stopped playing the rest of the night," Karlsson said. "It was the Devils' game after that.
"It's more [on] us. We let them play today and that is not how we should do it."
The Devils outshot the Golden Knights 37-15 after falling behind 3-0.
Keith Kinkaid made 14 saves in relief of Cory Schneider to earn the victory for the Devils (11-13-6), who have won twice in their past 10 games (2-4-4). Kinkaid entered at 9:36 of the first period after Schneider allowed three goals on seven shots.
"I just got to do my job and stop the puck and keep the score at bay," Kinkaid said. "We could have maybe had a few more goals out there. It was just an onslaught of shots on [Golden Knights goalie Marc-Andre Fleury]. It was a resilient effort tonight."
Fleury, who made 37 saves, lost for the third time in his past 13 games.
Karlsson scored on the power play at 3:57 of the second period to make it 4-1. He has four goals in his past three games and has a Vegas-leading 13.