Hossa's goal was the game-winner.
Goaltender Kari Lehtonen made 36 saves on 39 shots for the Thrashers, who got just the second win in their last six games.
Jordan Staal's goal in the second period cut Pittsburgh's deficit to 2-1 at the second intermission; Ryan Malone scored in the third period; and Sidney Crosby's goal with less than a minute left in regulation tied the game at 3-3, to send it into overtime.
Jocelyn Thibault gave up three goals on 32 shots for the Penguins, who have dropped three games in a row following a four-game winning streak.
On the power play, Atlanta was 2-for-7, courtesy of Kovalchuk and Koslov; while Pittsburgh went 0-for-7.