Derek Stepan and Taylor Hall also scored for the Coyotes (28-23-7), who scored three unanswered goals after going down 2-0 early. Antti Raanta made 26 saves.
Jake Evans and Brendan Gallagher scored for the Canadiens (27-24-7), who had won three in a row. Carey Price stopped 31 shots.
The Canadiens remain five points behind the idle Toronto Maple Leafs for third place in the Atlantic Division. The Leafs have two games in hand.
With Joel Armia in the box for hooking, Chychrun beat Price with a slap shot from the point for his 12th on the season. Arizona finished 1 for 6 with the man advantage.
The Coyotes had dropped seven of eight.
It was a dream start for the Canadiens, who scored twice on four shots on Raanta inside two minutes in an end-to-end first period.
It took just 51 seconds for Evans to find the back of the net. He got his first NHL goal when he jumped on Nick Cousins' rebound.
Gallagher made it 2-0 a minute later, scoring his fourth goal since returning from a head injury on Jan. 9. The winger's initial shot was blocked by defenseman Niklas Hjalmarsson, but he adeptly jumped on his own rebound.
Arizona got one back on Stepan's ninth of the year at 5:08 of the first. Price whiffed on the wrist shot.
A brawl at the end of the period, sparked by Oliver Ekman-Larsson cross-checking Nick Suzuki in the face, set the stage for a penalty-filled second period.
The Canadiens failed to score on a two-minute 5-on-3 power play. They finished the game 0 for 6 with the man advantage.
''We didn't play well enough to win,'' Gallagher said. ''There's no secret to that. We were pretty much sloppy. On the power play, our job is to score. And if you're not going to score, you need to build momentum.
''We didn't do that. We got the result we deserved.''
Despite Montreal's power-play opportunities, the Coyotes outshot the Canadiens 17-3 in the second period. Arizona was rewarded with Hall's equalizer at 16:36. Hall has 20 points in 23 games since joining Arizona.
''There's 5-on-3s and then there's full two-minute 5-on-3s with a bit of power play time hanging on at the end,'' Hall said. ''Good job by our killers. Raanta did a fantastic job.
''We forced the issue down the ice and disrupted their breakout. That was why they couldn't get pucks to the middle.''
Montreal's Phillip Danault took a puck to the mouth in the second period on a deflection. He left the game and did not return.
NOTES: Montreal D Shea Weber missed a third straight game with a lower-body injury. The captain will not travel for upcoming games in Boston and Pittsburgh. ... Super Bowl champion Laurent Duvernay-Tardif of the Kansas City Chiefs attended the game. ... Defenseman Brett Kulak played his 100th game with the Canadiens. ... Montreal beat Arizona 4-1 on Oct. 30.
Coyotes: At Toronto on Tuesday night.
Canadiens: At Boston on Wednesday night.
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