Cam Ward stopped 30 shots for the Hurricanes (3-2) and claimed his second one-goal win over Atlanta in seven days.
Sutter's game-winner came after he got behind Bryan Little, took a back-door feed from Erik Cole and buried the puck past Ondrej Pavelec. Sutter, Carolina's newest alternate captain, also scored a deflected power-play goal midway through the second.
Nik Antropov sent it into OT with 1:03 left, when he scored on a hard snap shot from the right circle after the Thrashers (0-5) pulled Pavalec for an extra attacker.
Sutter's first goal came with about 5 minutes left in the second when he redirected Joe Corvo's slap shot past Pavelec.
It made for a successful getaway day for the Hurricanes, who played a rare weekday matinee because they were leaving for Europe after the game. They wrap up the preseason Monday in Russia against SKA St. Petersburg before opening the regular season against Minnesota next week in Helsinki, Finland.
Pavelec made 28 saves for the Thrashers, who entered as one of only three winless teams this preseason. They've been outscored 14-7.