Koepke pushed the lead to 3-1 when he knocked in a rebound at the right post at 7:51.
Niederreiter extended it to 4-1, converting a 2-on-1 with Vladislav Namestnikov on a one-timer from the right circle at 16:42.
“I was just hoping [the puck] comes over," Niederreiter said. "I closed my eyes and shot the puck. I’m glad it went in.
"Obviously it was a big win for us. We came here to do that and we found a way to do that. It wasn’t pretty but we found a way and that’s the most important thing.”
Bratt pulled the Devils to within 4-2 with a power-play goal at 19:24 when he redirected a Hughes shot-pass at the top of the crease.
Hischier scored on a deflection from the slot with Allen pulled for an extra attacker at 18:14 of the third period for the 4-3 final.
"In the second period, we played a lot quicker and got out of our end, one or two passes, and were gone," Winnipeg coach Scott Arniel said. "Obviously we got some odd-man rushes off it and did a better job of forcing New Jersey back to their goal line to get pucks and the forecheck got working.
"In the third period, we knew the push was coming and, at the end of the day, that was a big two points for us."
NOTES: Niederreiter has 499 points (248 goals, 251 assists) in 1,020 NHL games ... New Jersey defenseman Dougie Hamilton had an assist to extend his point streak to eight games (one goal, nine assists). His eight-game assist streak is tied with Brian Rafalski (2006-07) and Barry Beck (1977-78) for the longest run by a defenseman in franchise history. ... Devils forward Cody Glass left in the second period after blocking a shot from Jets defenseman Josh Morrissey. He returned at the end of the period, but Keefe elected not to play him, and he did not return for the third. "He wanted to kind of give it a go there at the end of the second, so close to the intermission, but I didn't bother putting him out," Keefe said. "He did a test and it didn't get any better during the intermission, so he'll get further tests before we move on."