Christian Dvorak had his first two-goal game and Max Domi had a goal and an assist for Arizona (17-28-7), who scored three goals in the second period to erase a 2-0 deficit but couldn't hold the lead. Mike Smith made 35 saves.
"That was not even close," Coyotes coach Dave Tippett said. "We didn't have enough guys play well enough to win. When you don't deserve to win, you don't win games very often."
The bright spot was Domi, who had two points in his third game after missing almost two months following hand surgery.
"It was the first time since he's come back that he really had his skating legs," Tippett said. "We needed a few more players with skating legs."
Montreal made it 2-0 at 13:10 when Markov's shot deflected off the stick of Alex Goligoski and pinged in off the far post.
But the Coyotes rallied, scoring on the power play, even strength and shorthanded during a span of 2:21 of the second period to take the lead.
With the man-advantage, Dvorak took a Domi pass in stride and scored at 13:46. Domi tied the game playing 4-on-4 when he pushed the puck under Price and just over the line at 15:45. Twenty-two seconds later, Goligoski put a Jordan Martinook feed by Price to give Arizona a 3-2 lead.
"We've got to learn how to play with a lead," Domi said. "You have to take advantage and find a way to win. That last 20 minutes, we just came up short."
Pacioretty made it 3-3 with a power-play goal at 2:52 of the third, and he set up Alexander Radulov to put Montreal back in front 4-3 at 9:56.
But Dvorak won a race to the puck that Price tried to whack, and Price wound up kicking it into his own net at 15:08 to tie the score.
"We talked here in the locker room how we needed to get a win no matter what," Galchenyuk said. "It wasn't the perfect game but we got the win."
