Koivu scored twice, including a power-play goal 1:40 into overtime as Montreal completed a late comeback for a 4-3 win over the Edmonton Oilers on Tuesday night.
The Canadiens captain deflected Mathieu Schneider's point shot past Dwayne Roloson for the winner after he tied it with 4:35 left in regulation on a goal that required a lengthy video review.
Referee David Banfield pointed to the net repeatedly when the puck deflected up off the stick of Oilers defenseman Steve Staios on Andrei Kostitsyn's centering pass and caromed in off Koivu, who drove hard to the net.
"I knew that it hit me on my chest and I knew that I didn't intentionally try to grab the puck or anything," Koivu said. "I was pretty confident that it was a good goal, but you never know. It was a pretty long period that we had to wait there, and it was a relief."
Montreal killed off a penalty late in regulation, and Koivu scored his second of the game after Patrick O'Sullivan was sent off for slashing 50 seconds into overtime.
"Overall, we played a pretty good game," Oilers coach Craig MacTavish said. "There were a lot of positive signs in the game, but it stings when you lose a game like that where you're up with under six minutes to go and you don't get two points. We probably could have negotiated one point from Montreal and taken two."
Roloson made 22 saves in his 22nd consecutive start for the Oilers, extending his team-record streak.
Tanguay was sent off with 2:10 remaining in the third for tripping Shawn Horcoff.
Price stopped 29 shots in his fourth straight start for the Canadiens, who won 3-1 in Dallas on Sunday in their final game under Carbonneau.
Montreal, which lost 10 of 13 in late January and February, had dropped the first two games of its three-game road trip in the wake of a four-game winning run before the trade deadline.
"We made a few changes in our game," Koivu said. "We're trying to keep the puck a bit more and try to make clean plays and then simplify when we get in the neutral zone."
Gainey also took over behind the bench in Montreal after he fired Claude Julien midway through the 2005-06 season.
The Canadiens went 23-15-3 under Gainey through the end of the regular season before losing in the first round of the playoffs.
"He really brings a calming demeanor into the locker room," Price said. "He's kind of like the more strong, silent type, whenever he does say something you really have to listen because you know it's going to be something important."
Gagner scored his 10th goal off a goal-mouth scramble 18:32 into the second to restore Edmonton's one-goal lead.
Cogliano scored the first of Edmonton's three goals in the second to tie it at 1 midway through the period. Souray put the Oilers ahead for the first time at 14:36.
Outshot 12-0 in the second following Souray's goal, Montreal tied it at 2 on its second shot of the period when Metropolit scored his first goal in six games since the Canadiens picked him up on waivers from Philadelphia on Feb. 27.
Metropolit scored his fifth goal at 16:26.
Tanguay opened the scoring with his 11th goal 8:12 in. It was his first goal in three games since returning from a shoulder injury that sidelined him for two months.
Tanguay went to the dressing room briefly early in the second after he was cross-checked in the face by Souray in front of the Oilers net on a play that went unpenalized.
"I know they all make mistakes, but that's a game-changing event," said Tanguay, who had four stitches above his upper lip. "It's a four-minute penalty, and that can change a game around, especially with the game meaning so much to both sides at this time of the year. It's a call that the referees can't miss."
Gainey entered with a 188-205-63 coaching record in 456 regular-season games over the course of six seasons with the Minnesota-Dallas franchise and Montreal. ... MacTavish said Souray's condition would be evaluated Wednesday, though he didn't expect the defenseman would be out long. Souray wanted to return in the third period, but was kept out for precautionary reasons. ... Tanguay missed every game in January and February as well as the Canadiens' first game this month, a 5-1 loss in Buffalo on March 4. ... Montreal reached 4,000 NHL points on home ice.
Copyright by STATS LLC and The Associated Press
Arena: Bell Centre Location: Montreal, Quebec
Referees: David Banfield, Dan O'Halloran Linesmen: Michel Cormier, Steve Miller
Attendance: 21,273 (100.0% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Alex Tanguay (11)
Assists: Mathieu Schneider, Josh Gorges
Andrew Cogliano (16)
Assists: Steve Staios, Sheldon Souray
Sheldon Souray (18)
Assists: Ales Hemsky, Patrick O'Sullivan
Glen Metropolit (5)
Assist: Mathieu Dandenault
Sam Gagner (10)
Assists: Marc-Antoine Pouliot, Denis Grebeshkov
Saku Koivu (12)
Assists: Andrei Kostitsyn, Alex Tanguay
Saku Koivu (13)
(Power Play) Assists: Mathieu Schneider, Andrei Markov