The Hawks threw away a chance to end the game in regulation, but eventually prevailed to win their second in a row and deal the Pistons their first loss in three games.
A 3-pointer by Teague put Atlanta in front for good with 2:30 to play in the second extra session and his layup with 2 minutes left pushed the Hawks ahead by five points.
Atlanta had a six-point lead with 90 seconds left in the fourth quarter, but the Pistons rallied to go in front by one on a 3-pointer by Austin Daye with 4 seconds to go.
The Hawks' Al Horford was then fouled with 1.8 seconds remaining and he made the first free throw to tie the game. His second attempt, however, bounced away to force the first overtime.
Neither team could score in the final 51 seconds of the first extra period after Teague tied it with two free throws.
The Pistons were led by Will Bynum with 31 points.