Hedberg made 28 saves, 21 in the first two periods. Kovalchuk scored a goal in regulation and Marian Hossa had two assists for the Thrashers (3-2-0).
Ellis made 13 saves in the third period and another in overtime for Nashville (3-1-1) after replacing regular goalie Chris Mason, who stopped 14 shots.
The Predators had the first eight shots and took a 1-0 lead on J.P. Dumont's unassisted goal at 13:58 of the first period. Atlanta tied it on Kovalchuk's power-play shot with 8 seconds remaining in the first period.
It was the first goal of the preseason for Kovalchuk, who was second on the Thrashers with 42 last season.