Kovalchuk took a pass from Marian Hossa, sped across the blue-line and beat Mike Dunham with a shot just inside the far post.
Kovalchuk, who has 26 goals on the season, also scored during a first-period spurt that led the Islanders to yank their starting goalie, Rick DiPietro, before the game was six minutes old.
But New York kept coming back, rallying to tie the game at three in the second period and bouncing back again to force overtime after Hossa put Atlanta back in front with a goal 19 seconds into the third.
In fact, the Islanders were pressing hard for the winning goal in the final minute of regulation. But Kari Lehtonen turned away several good chances, and the Thrashers won it in overtime.