Kozlov beat Capitals goaltender Brent Johnson to open the shootout. Thrashers goalie Kari Lehtonen then stopped Alexander Semin before Hossa beat Johnson. Lehtonen blocked Alex Ovechkin's shot to finish off the 2-0 shootout.
Hossa, who scored earlier, got a tying goal with 33 seconds left in regulation to send it into overtime. His seventh goal came when he took a pass from Ilya Kovalchuk and knocked it over the right shoulder of Johnson.
Ovechkin scored in the second period to give Washington a 3-2 lead at 15:48.
The Thrashers jumped in front with a power-play goal in the first period. Hossa redirected a shot by Kovalchuk past Johnson, making only his second start, at the 5:02 mark. It came only 14 seconds after the Capitals' Kris Beech was penalized for holding.
Kozlov knocked in his third goal from in close on a slick pass from Hossa to give the Thrashers a 2-0 lead 1:04 into the second period.
Washington bounced back after Kozlov's goal, scoring three times in the second period. Dainius Zubrus converted his third at 3:39 and Semin picked up his seventh goal on a power play at 14:51 off a faceoff from the left circle.
Less than a minute later, Ovechkin, who as a rookie a year ago scored 52 goals, got his third goal of the season, scoring from up close to put the Capitals in front.
Atlanta, which had won four of five, outshot Washington 39-25 in regulation, including 1-for-19 in the final period with Johnson making several outstanding stops. The Thrashers were only 1-for-7 on power plays and had a chance to score early in the third period when they had a two-man advantage for 1:15, but were stymied by the Capitals' defenders.
Notes: Lehtonen is now 5-0 against Washington. ... Semin and Chris Clark have scored a point in all six of the Capitals games. Clark got an assist Zubrus' second period goal. ... Lehtonen also gave up three goals in one period (first) in a 4-3 loss to Carolina on Oct. 13.