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Alex Kovalev had three assists and Chris Higgins added two assists for the Canadiens, who are 6-0-3 against the Coyotes since 1998. Montreal's Carey Price had 37 saves, including one on a breakaway by Shane Doan.
Phoenix took a 2-1 lead at 3:54 of the third period when Zbynek Michalek picked up a turnover along the left boards and fired a shot from the faceoff circle, beating Price under the crossbar.
The Canadiens tied it three minutes later on Koivu's power-play goal after Joel Perrault was called for hooking.
Montreal went ahead on Markov's power-play goal at 10:07. Markov took a feed from Kovalev in the slot and beat Phoenix goal Mikael Tellqvist. Markov also had an assist.
Kostitsyn clinched it with a goal on a 2-on-0 breakaway at 18:19.
The Coyotes started a five-game homestand, but that might be a mixed blessing for a team that entered 14-15-2 in Jobing.com Arena.
Phoenix, chasing its first playoff berth since 2002, entered in 10th place in the Western Conference.
Montreal took a 1-0 lead at 6:03 of the second period when Tomas Plekanec took a pass near the goal line, wheeled and fired a low wrist shot past Mikael Tellqvist. It was Plekanec's 27th goal, and Kovalev picked up his 500th career assist.
Four minutes later, Phoenix tied it at 1-1 after Montreal's Maxim Lapierre drew a 4-minute double-minor for high-sticking. Shane Doan tipped Ed Jovanovski's blast from the blue line.
Lapierre was still in the box when Doan sailed in on a breakaway a minute later, but Price blocked his point-blank shot.
Notes: Phoenix's last victory over Montreal came on Dec. 9, 1998. Montreal was 5-0-3 against the Coyotes since then. ... The Canadiens had not played in the desert since March 5, 2004. ... Phoenix had 30 shots through two periods
-- three fewer than it had mustered in its previous two games.
[Associated Press; By ANDREW BAGNATO]
Copyright 2008 The Associated Press. All rights reserved. This
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