Kozlov, Ilya Kovalchuk and Todd White each had a goal and an assist in regulation for the Thrashers, who have won seven of their last nine games.
Hedberg improved Atlanta's shootout record to 6-1, stopping attempts by Steven Stamkos, Vinny Lecavalier and Martin St. Louis. At the other end, Karri Ramo shut down Bryan Little before Kozlov netted the winner.
Tampa Bay, 3-10 in shootouts, has lost six consecutive home games and nine of its last 10.
Kozlov opened the scoring 6:12 into the game. White made it 2-0 during a two-man advantage late in the first period, when the Thrashers took 19 shots.
White has at least a point in nine straight games. He has a career-high 66 points, including three goals and two assists in five games against Tampa Bay.
Early in the second, Stamkos beat Hedberg twice in 1:22 to pull the Lightning even. The top overall pick in the 2008 draft has eight goals and five assists in his last 12 games.
St. Louis helped set up Stamkos' first goal, extending his point streak to eight games. Tampa Bay's leading scorer has five goals and six assists in that span.
Kovalchuk put the Thrashers ahead 3-2 on a power play at 12:52 of the second, but Vinny Prospal tied it less than 3 minutes later.
Ramo finished with 33 saves, and Hedberg stopped 26 shots.
NOTES: The Thrashers have won nine of the last 11 games in the series. They have won four of their last five in Tampa. ... Kozlov has 38 points on the power play, second to NHL leader Alex Ovechkin. ... The Lightning scratched D Marek Malik and D Cory Murphy because of foot injuries. ... Thrashers C Jim Slater and D Boris Valabik were sidelined by shoulder problems. ... Atlanta has allowed 15 goals in its last three games.
Atlanta Thrashers goalie Johan Hedberg, of Sweden, has the net land on him during the second period of a hockey game against the Tampa Bay Lightning Saturday, March 21, 2009, in Tampa, Fla. (AP Photo/Mike Carlson)
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