Perrin faked to his right before pulling the puck back and shooting past Thibault to the left side of the net, ending the Sabres' three-game winning streak.
Thrashers goalie Johan Hedberg stopped shots by Ales Kotalik, Derek Roy, Clarke MacArthur and Thomas Vanek in the shootout.
Mark Recchi, Bobby Holik, Jim Slater and Tobias Enstrom scored in regulation for Atlanta. Jochen Hecht, Jason Pominville, Daniel Paille and Dmitri Kalinin had the Sabres' goals.
Buffalo led 4-3 but was hurt by two penalties in the last minute of regulation.
MacArthur was called for hooking with 56.4 seconds left, and Tony Lydman drew another penalty for delay of game with 19 seconds left.
With Hedberg pulled, the Thrashers had a three-man advantage that paid off quickly as Enstrom found the net with a long shot from beyond the right circle with 17 seconds left, forcing the overtime.
Buffalo's Ryan Miller gave up two first-period goals but shut out the Thrashers in the second period. Miller stopped 18 shots through two periods but suffered a leg injury in the second period. He left with a 3-2 lead and was replaced by Thibault to start the third period.
Thibault gave up a game-tying goal to Slater 2:25 into the third period, but Kalinin answered with a power-play wrist shot for the Sabres less than 3 minutes later for a 4-3 Buffalo lead.
Hedberg, looking to atone for giving up seven goals in a 10-1 loss to the Sabres in Buffalo on Jan. 18, couldn't hold a 2-0 lead after one period. The Sabres rallied with second-period goals by Hecht, Pominville and Paille.
Hedberg and the Thrashers played strong defense early in their first of at least two games without Ilya Kovalchuk, who hurt his right knee when hit by Pittsburgh's Jarkko Ruutu Wednesday night.
The MRI on Kovalchuk's right knee was negative. He also will miss Atlanta's game at Washington on Saturday.
The Thrashers had another injury scare when their other All-Star, Marian Hossa, was sent crashing hard into the boards behind the Buffalo net by Jaroslav Spacek early in the third period. Hossa left the game but returned later in the period.