Jesperi Kotkaniemi gave Montreal a 1-0 lead at 3:30 of the first period, redirecting a shot from Jeff Petry.
"I just like the big games," Kotkaniemi said. "Every game matters, so trying to do your best and you're trying hard enough, every once in a while you get the puck in the net."
Staal scored his first of the Stanley Cup Playoffs to make it 2-0 at 5:10 when he shot into an open net after receiving a cross-crease pass from Perry.
"Playoffs are about adapting and getting to know your opponent and being ready for your opponent and knowing who is on the ice," Perry said. "They have guys over there that can break a game open with speed and skill and guys that can score. We know who's on the ice, we just have to be aware of that. We just have to take care of ourselves and we're going to be OK."
Adam Lowry cut it to 2-1 at 11:52, stealing the puck from Erik Gustafsson and scoring shorthanded on a breakaway.
"I think we knew they were going to come out, they were going to have their legs right away [with] them coming off that emotional game in Toronto and, you know, being sharp," Lowry said. "I think we just lacked a little sharpness early. A couple of mental mistakes. Dug ourselves a hole but, you know, [we] didn't quit."
Nick Suzuki pushed the lead to 3-1 at 17:14, toe-dragging Hellebuyck on a 2-on-1 before tucking a shot inside the left post.
Derek Forbort cut it to 3-2 at 9:22 of the third period, and Brendan Gallagher scored a power-play goal on the rebound of Shea Weber's shot to make it 4-2 at 11:04.
With Hellebuyck pulled for the extra skater, Kyle Connor made it 4-3 with a one-timer at 18:18.
"I thought [the Canadiens] were real good. They looked like they were right in their rhythm. We weren't as good as we have been," Maurice said. "We didn't come to the rink to play regular-season hockey. We understand. We're excited about the challenge we have here. We can get better, we know that. They played a [heck] of a game."
NOTES: Jets forward Paul Stastny did not play for undisclosed reasons and was replaced by Dominic Toninato, who had one hit in 5:33 of ice time. Maurice said Stastny is day to day. … Winnipeg defenseman Dylan DeMelo left the game after sustaining a lower-body injury on the opening shift of the game. Maurice said he will have a better idea of his status Thursday.