It was Laine's eighth career hat trick and gave the Jets points in four straight games after a five-game losing streak.
"Gotta find new ways to score. Haven't scored that many goals from my spot," said Laine, who has eight goals in his past eight games. "So just try to change it up a little bit, go to different spots and maybe get those kind of dirty goals and tips, that kind of stuff. So that was a good day."
It was his effort backchecking before his third goal that made him feel extra good.
"Yeah, well, I started it. It was my pass and it went the other way," Laine, 22, said. "You've always got to backcheck, and it doesn't matter what the score is. That's going to look stupid if you just stand there and not try.
"I'm proud of that, even if after that the result's not a goal. It doesn't matter. I still take a lot of pride from that play. Gotta keep playing like that."
"I thought there was a lot of emotion during 5-on-5," Pageau said. "But that is where we need to control our emotions and stick with our game plan and don't give them the chance to go on their power play.
"That's where they have their success, and that's where they killed us tonight."
Laurent Brossoit made 25 saves for Winnipeg, which was beginning a season-long, six-game homestand.
Winnipeg led 1-0 after the first period and 4-2 after the second.
Pionk scored his sixth goal of the season when a one-timer from the point went past a screened Anderson at 8:48 of the first period.
With just under six minutes remaining in the first period, Senators forward Brady Tkachuk crushed Pionk against the edge of the boards between the two benches. Jets captain Blake Wheeler pulled Tkachuk away and the two started throwing punches.
Pageau netted his 21st goal of the season 25 seconds into the second period with a backhand swipe at the puck went by Brossoit to make it 1-1.
Winnipeg regained the lead at 7:49 just seconds after Nikolaj Ehlers exited the penalty box for tripping. Ehlers got the puck, sent a long pass to Connor and he beat Anderson for his team-leading 26th goal of the season.
The helper extended Ehlers' point streak to five games, with one goal and four assists during that span.
Laine put away his 21st goal of the season by banging in a rebound on the power play at 17:04 to make it 3-1, but then a costly Winnipeg turnover cut the deficit to one again.
Brossoit went behind the net and tapped the puck to Pionk beside him, but the defenceman passed the puck out directly to Boedker, who easily fired it into the net at 18:29 for his first goal of the season in his 10th game.
Laine made it 4-2 when he deflected in a Pionk shot with 12 seconds remaining in the middle frame and upped the score once more at 8:43 of the third.
It was Laine's first hat trick since he had a five-goal performance on Nov. 24, 2018 against St. Louis. He also surpassed last season's point total (50) with the third goal. He sits at 23 goals and 51 points in his fourth NHL season.
Senators: visit Avalanche on Tuesday.
Jets: hosts Blackhawks on Sunday.