Damon Severson had a goal and two assists, and Nikita Gusev had three assists for the Devils. Mackenzie Blackwood made 32 saves for his third win in four games.
"Our goalie was huge as well, along with the penalty kill, and the game turned as we went on," Severson said of Blackwood and the Devils penalty kill, which went 7-for-7. "That was a huge win right before the break."
Zack Smith scored for Chicago (15-17-6), which was denied a third straight win entering the Christmas break. Corey Crawford allowed four goals on 20 shots in 30:40 before he was pulled for Robin Lehner, who made 21 saves in relief.
"Just an embarrassment," Blackhawks captain Jonathan Toews said. "I don't know if that's an emotion, but that's what we're all feeling and what we should be feeling at this moment."
Smith gave the Blackhawks a 1-0 lead on a wraparound at 3:21 of the first period.
Jack Hughes ended his 18-game goal drought at 14:56 of the first when he lifted a far-side shot over Crawford's blocker to tie it 1-1.
Severson scored his first goal in 11 games when he skated around Toews at the left face-off circle and put a shot between Crawford's legs for a 3-1 lead at 7:22 of the second.
"I thought I had it, but I mean [it] just seemed like nothing was going right," said Crawford, who has lost his past four starts.
Hayden snapped a wrist shot from the left circle at 10:40 of the second to make it 5-1.
New Jersey outshot Chicago 23-8 in the second period.
Blake Coleman scored 18 seconds into the third to make it 6-1, and Pavel Zacha scored at 18:21 for the 7-1 final.
"Yeah, it's a missed opportunity," Blackhawks coach Jeremy Colliton said. "We should be embarrassed. If we're going to make progress, you've got to give yourself a chance to make progress. I don't think we did that tonight."
Brothers Jesper Boqvist and Adam Boqvist played their first NHL game against each other, but each left in the first period with injuries. Jesper Boqvist, a Devils forward, left after 1:42 and was held out for precautionary reasons, Devils interim coach Alain Nasreddine said; Boqvist, a Blackhawks defenseman, exited after 2:42, with a "minor" right shoulder injury, Colliton said.