In their two previous meetings this season, the teams also went to the shootout tied at 2. Atlanta came out on top in those games, but the Canadiens survived this one for their fourth win in five games.
"You're never too confident in a shootout," Koivu said.
Huet was eager to make up for his blunder in the third, which squandered a 2-1 lead.
As a Montreal power play was ending, he ventured into the corner to play the puck. But Huet froze when trying to decide whether to give it up to teammate Roman Hamrlik or simply clear the zone.
Eric Perrin sneaked in to poke the puck away, and it skidded right in front of the unguarded net to Pascal Dupuis. He buried it for his ninth goal with 10:58 left in regulation.
But the Thrashers went 0-for-2 in the shootout. Slava Kozlov shot wide and Marian Hossa's attempt to go five-hole was blocked by Huet, who squeezed his pads together in the nick of time.
"It was a good chance to redeem myself," the goalie said.
Huet wasn't too thrilled about Atlanta's first goal, either. Hossa attempted a pass across the crease, but the puck struck Huet in the arm and wound up behind him to give Atlanta a 1-0 lead on the power play late in the first.
"It was a good opportunity for me to really step up and make sure we didn't get beaten in the shootout," Huet said. "I definitely felt I owed the team something after the two goals."
The Thrashers lost for only the third time in 13 games that went to extra time.
"We've been pretty good in those situations," said coach Don Waddell, whose team squandered a chance to get back to .500 overall. "We didn't get that extra point, but we got one."
Before his shootout goal, Kostitsyn tied the game at 1 early in the second period with a brilliant rush to the net.
The 20-year-old Russian took a pass near center ice, cut to his left to avoid Perrin, shifted back to his right to get around defenseman Ken Klee and flipped a shot past Kari Lehtonen for his 12th goal of the season.
"Andrei was very good tonight," Huet said.
Michael Ryder gave the Canadiens their first lead with 13:58 left in the third, finishing off a nifty bit of passing.
Skating along the boards, Bryan Smolinski worked the puck over to Guillaume Latendresse, who relayed it to Ryder cutting in front. He jabbed at it with his stick, managing to deflect a wobbly shot over Lehtonen's left shoulder.
The Canadiens are heading home after a four-game road trip, though they might want to stay away. They improved to 16-10 on the road, compared with an 8-12 mark in Montreal.
"We seem to be a lot more comfortable on the road," coach Guy Carbonneau said. "We still have a young team, and on the road you have less distractions. Everyone kind of sticks together and that makes it a lot easier."
Even with the loss, the Thrashers earned a point to claim sole possession of first place in the weak Southeast Division. Carolina dropped a point behind with a 5-1 loss to Ottawa.
Every team in the division is below .500.
"Wins are hard to come by," Waddell said. "If we only give up two goals, we're going to win most of those games. Unfortunately, we just couldn't get any offense going."
Atlanta defenseman Tobias Enstrom took a hard hit behind the net from Sergei Kostitsyn, knocking off his helmet. Enstrom skated slowly to the locker room but returned to the game after getting checked out. ... Montreal improved to 7-5 in shootouts, while the Thrashers dropped to 5-3. ... Hossa reached 20 goals for the eighth straight season. ... Montreal was 0-for-4 on the power play. ... Lehtonen made 27 saves, while Huet had 22.
Copyright by STATS LLC and The Associated Press
Arena: Philips Arena Location: Atlanta, Georgia
Referees: Gord Dwyer, Dave Jackson Linesmen: Jean Morin, Jonny Murray
Attendance: 16,181 (87.3% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Marian Hossa (20)
(Power Play) Assists: Mark Recchi, Eric Perrin
Andrei Kostitsyn (12)
Assists: Tomas Plekanec, Andrei Markov
Michael Ryder (7)
Assists: Guillaume Latendresse, Bryan Smolinski
Pascal Dupuis (9)
Assist: Eric Perrin
No scoring this period
Slava Kozlov shootout attempt against Cristobal Huet with a Over Net results in a MISS.
Andrei Kostitsyn shootout attempt against Kari Lehtonen with a Wristshot results in a GOAL.
Marian Hossa shootout attempt against Cristobal Huet with a Wristshot results in a SAVE.
Saku Koivu shootout attempt against Kari Lehtonen with a Snapshot results in a GOAL.