Atlanta, coming off a 3-2 victory over Eastern Conference-leading Ottawa on Thursday, has won three straight after losing four in a row and eight of 11.
"I thought tonight we started out a little slower than in the other two, but as the game wore on, it got better," coach Don Waddell said after the Thrashers improved to 4-1 in shootouts. "We knew it would be tougher because they play a very patient game."
The Canadiens dropped to 2-5 shootouts despite beginning the game tied for fifth in the NHL with 23 road points. They had won four straight and five of six away from Montreal.
"There were a lot of loose pucks near the net," Canadiens center Saku Koivu said. "We just didn't get the puck in the net."
Kozlov gave the Thrashers a 1-0 lead with his 15th goal. Hossa had two assists.
Lehtonen, who made 32 saves in regulation and three in overtime, stopped the two attempts he faced in the shootout.
"I think Lehtonen played a strong game," Montreal coach Guy Carbonneau said. "He was the one who saved the game for them."
Montreal forced a 1-1 tie on Alex Kovalev's 15th goal with 11:40 remaining.
The Thrashers took a 2-1 lead on Eric Perrin's seventh goal. Perrin, who assisted on Kozlov's goal, scored the power-play tally with 1:07 left in the second.
Hossa was credited with his 21st assist, and second of the game, after drawing two defenders toward the right side goal line and passing across the top of the crease for Perrin's tap-in.
"I couldn't see the puck," Perrin said. "I just kept my stick on the ice."
Montreal's Cristobal Huet, 3-1 in his last four starts against Atlanta, stopped 26 shots. He had missed seven straight games due to a pulled groin before winning 5-2 at Washington on Thursday.
Huet praised Hossa after the Thrashers forward sent the puck between the goalie's legs in the shootout.
"I was surprised he shot so quick," Huet said. "I'm not very confident in our shootouts, but it was nice to get one more point [in the standings]."
Kozlov, who gave the Thrashers a 1-0 lead 4:31 into the first, used a wrister to shoot from the boards outside the left circle and sent the puck over Huet's left shoulder.
Nearly eight minutes later, Sergei Kostitsyn made it 1-1 with a wrist shot from the right side of the slot.
Lehtonen is 5-2 since returning from a groin injury that sidelined him for 16 games. He stopped shootout attempts from Kostitsyn and Koivu.
"The shootout is something we try to work on," Carbonneau said, "but it's not something we can spend too much time on."
Kozlov, who scored his 15th career goal against Montreal, beat Huet in the shootout by faking a straightaway wrister from the slot and then swinging left and beating the goalie to the stick side.
Kostitsyn went next but failed in his attempt to try the same move against Lehtonen, who dove to his right and kept the puck out of the crease.
Lehtonen felt fortunate that Koivu didn't beat him.
"I had faced Saku Koivu in Europe," Lehtonen said. "I thought he was going the other way. I tried to poke the puck and missed. I guess he added another move since I have seen him. It was luck, I guess."
In the last 12 games, Atlanta has killed 39 of 42 penalties. In their first 17 games this season, the Thrashers gave up 28 of 85. ... Montreal began the game with the NHL's best power-play unit at 28.4 percent. ... The Canadiens have lost three straight to Atlanta. They dropped to 9-4-2 in the overall series at Philips Arena and 19-9-2 overall against the Thrashers.
Copyright by STATS LLC and The Associated Press
Arena: Philips Arena Location: Atlanta, Georgia
Referees: Don Koharski, Chris Rooney Linesmen: Darren Gibbs, Thor Nelson
Attendance: 17,153 (92.5% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Slava Kozlov (10)
Assists: Eric Perrin, Marian Hossa
Sergei Kostitsyn (2)
Assists: Saku Koivu, Mike Komisarek
Eric Perrin (7)
(Power Play) Assists: Marian Hossa, Mark Recchi
Alex Kovalev (15)
Assists: Roman Hamrlik, Andrei Kostitsyn
No scoring this period
Marian Hossa shootout attempt against Cristobal Huet with a Snapshot results in a GOAL.
Saku Koivu shootout attempt against Kari Lehtonen with a Wide of Net results in a MISS.
Slava Kozlov shootout attempt against Cristobal Huet with a Snapshot results in a GOAL.
Sergei Kostitsyn shootout attempt against Kari Lehtonen with a Snapshot results in a SAVE.