Paul Byron scored the go-ahead goal in the third period, and Shea Weber scored on the power play for the Canadiens (18-13-5), who had lost two in a row.
Oliver Ekman-Larsson scored, and Darcy Kuemper made 26 saves for the Coyotes (14-18-2), who have scored two goals while losing their last three games and have lost seven of eight.
Byron made it 2-1 at 1:16 off Phillip Danault's backhand pass to the left post. It was Byron's fourth goal in eight games.
"[He] is one of those guys who's always going to give you a great effort. He was at the right place at the right time," Canadiens coach Claude Julien said. "Most of the goals [now] are being scored in front of the net. He was in the right place, Phil wins the battle along the wall, we keep the puck in and we end up scoring a big goal."
Weber scored 22 seconds into the second period to make it 1-0 on a one-timer from the point following Domi's face-off win. Weber's goal was his fifth in 11 games in December and came after Nick Cousins was called for tripping as the first period ended.
Ekman-Larsson tied it at 1-1 on the power play at 6:49 with a wrist shot from the top of the right circle as Christian Fischer screened Price.
"I mean, Carey Price is world class, and he made some great saves," Coyotes coach Rick Tocchet said. "We missed the net a few times when we had people in front, and we have to hit those nets."
Montreal won the second half of a back-to-back after losing to the Colorado Avalanche 2-1 on Wednesday, when Price made 24 saves.
"I thought guys dug in," Weber said. "I thought at times [the Coyotes] were fresher on the back-to-back, but guys found a way. But it wasn't easy."
Julien played Price in back-to-back games after Price played well against the Avalanche.
"I would tell you that Carey has been playing pretty good," Julien said. "When a goaltender is hot, a lot of times you like to ride them."
Neither Domi nor Coyotes forward Alex Galchenyuk had a point. The two were traded for each other June 15 and were playing against their former team for the first time.
Domi took two shots in 18:17 and missed an attempt at an empty-net goal in the final minute. Galchenyuk had two shots in 16:30.
"That was fun. I have so many memories here, and it was emotional," Domi said. "I was here for only three years but it seems like 20."
