Trevor Lewis made it 2-1 at 8:03 when he drove to the net and, despite being covered by Canadiens defenseman Victor Mete, got his stick down to tip in Neal Pionk's pass from the right side.
"He made an awesome pass and I got a piece of it and it went in," Lewis said.
Nikolaj Ehlers drove the left side and shot past Allen's glove at 13:07 for a 3-1 lead.
"I don't think it was a lack of effort," Weber said. "I just think we made some mistakes, and the mistakes were big, and obviously those are chances that we can't give up."
Byron took advantage of an unusual bounce on Jeff Petry's dump in along the boards to make it 3-2 at 15:14 of the second period. He shot into an open net when the puck caromed to him after Hellebuyck left the crease to play it.
"I get a little bit angry because I have no control over that kind of a bounce," Hellebuyck said. "I just move on, and that's kind of the story of this year, those bad] bounces. I just can't really afford to give up another one, so that becomes my main focus, shutting it down."
Andrew Copp scored into an empty net with 38 seconds remaining for the 4-2 final.
NOTES: Ehlers leads Winnipeg with five game-winning goals. ... Jets forward Blake Wheeler did not travel because of a head injury he sustained when he was elbowed by Ottawa Senators forward Brady Tkachuk during a 4-3 win Monday. "He has been diagnosed with a concussion so he's in the protocol now," Jets coach Paul Maurice said. ... Canadiens goalie Carey Price missed his second game because of a lower-body injury and will not be available for the end of the two-game set here Saturday.
NHL.com staff writer Tim Campbell contributed to this report
[Video: Hellebuyck stops 36 shots in a 4-2 win over Canadiens