After the first three shooters in the extra segment failed to score, Kozlov fired a wrist shot past the stick side of Tampa Bay netminder Karri Ramo.
Tobias Enstrom had two assists and White had one, while Johan Hedberg was credited with 26 saves on 29 shots for the Thrashers, who snapped a two-game skid and rose to 6-1 in shootouts.
Tampa Bay got Steven Stamkos' 17th and 18th goals and Vaclav Prospal's 18th (power play), all in the second period. Lukas Krajicek had two assists, while Ramo was beaten three times on 36 shots in the Lightning's third straight loss.
Tampa Bay sank to 3-10 in shootouts.