The Hurricanes had won nine straight at the Bell Centre -- including three playoff victories in 2006 -- since the lockout that canceled the 2004-05 season. Montreal hadn't beaten Carolina on home ice since recording a 1-0 overtime victory on Feb. 28, 2004.
"We've had some problems against that team, so we'll take these two points," Koivu said. "Sometimes in a long season you don't play your best or you don't play the way you were hoping, nights like that when you can get two points, that's the bottom line."
Alex Tanguay scored on a penalty shot in the final seconds of the first period and Price stopped 31 shots for Montreal, which closed out a 4-1 homestand despite falling 6-4 to Anaheim on Saturday in the team's first regulation loss of the season.
The Canadiens' power-play unit of Koivu, Alexei Kovalev, Robert Lang and Andrei Markov pressed hard in overtime for the winning goal but couldn't beat Ward. He made a huge stop on Lang after Staal drew a tripping penalty 2:35 into the extra session.
Kovalev scored an unassisted goal 15:56 in to make it 1-1.
With the Canadiens working on a 5-on-3 advantage late in the period, Carolina defenseman Dennis Seidenberg closed his hand on the puck in the crease. Tanguay was awarded a penalty shot for the infraction, and he beat Ward with 10.1 seconds remaining for his sixth goal of the season to put Montreal up 2-1.
"He's a good goalie," Tanguay said of Ward. "He moves well and obviously at the end of the period you don't want to mess around with the puck in the slot because the ice gets pretty chippy out there. You just want to make sure you get a good shot, and I got it over the pad and fortunately for us it went in."
Cullen drew Carolina even with his fourth goal, scored 40 seconds into the third. Cullen put a rebound of Seidenberg's shot under Price's left pad.
"It was a shot and it popped out of my glove and it went to the side of the net," Price said. "I thought it was going to get chopped away, but it ended up getting chopped toward the net. I was in the middle of getting up and it popped through my legs and I couldn't get back down again."
Price got an automatic delay of game penalty with 2:01 remaining in the third when he played the puck aggressively and cleared it over the glass.
"The guys did a great job -- I think they only got one or two shots, maybe, right at the end," Price said. "I had to go to the bench and thank all the guys for that one."
Anton Babchuk scored his first of the season midway through the first for Carolina, which fell to 2-1-2 on its longest road trip of the season, a six-game stretch that will end Thursday in St. Louis.
"I thought we did a good job and that's what's frustrating about the shootout, now everybody's saddled with a loss and it's like you buried your dead cat in the locker room -- nobody's talking," Hurricanes coach Peter Laviolette said.
The Canadiens honored former coaches Dick Irvin, Toe Blake and Scotty Bowman before the game as part of the team's 100th season celebrations.
Bowman, currently a senior adviser with the Chicago Blackhawks, was on hand for the ceremonial faceoff along with broadcaster Dick Irvin Jr. and Blake's son, Bruce.
Andrei Kostitsyn returned to Montreal's lineup after missing two games because of a concussion.
Sutter is out indefinitely with a concussion from an open-ice hit by the Islanders' Doug Weight. Pitkanen tore the meniscus in his knee and is expected to be sidelined for a month.
Babchuk's shot from the right point off a faceoff in the Canadiens zone struck Koivu's stick and deflected past Price for a power-play goal 11:04 in.
- The Hurricanes have not lost in regulation in Montreal in almost five years. Carolina is 6-0-2 in the regular season at the Bell Centre since a 3-1 loss to the Canadiens on Dec. 6, 2003.
- Irvin, Blake and Bowman coached the Canadiens to a total of 16 Stanley Cup championships between 1940 and 1979.
- D Patrice Brisebois, RW Georges Laraque and LW Steve Begin were healthy scratches for Montreal.