The Canadiens have not lost to Phoenix in 13 seasons.
Montreal's regulation goals came in the second period from Brian Gionta and Travis Moen, giving the Canadiens a 2-1 lead going into the third. Raffi Torres tied it on a wrist shot 5:50 into the final period.
Carey Price made 32 saves for Montreal, which had lost two in a row. Mike Smith gave up three goals on 36 shots.
The last win for the Coyotes over Montreal came Dec. 9, 1998. Thursday's game was their only scheduled meeting this season.