The Eastern Conference-leading Sabres, winning for just the fifth time in their last 12 games, scored all three of their regulation goals on the power play. They took the shootout 2-0, getting goals from Briere and Ales Kotalik.
Ryan Miller stopped Slava Kozlov and Marian Hossa in the shootout.
Scott Mellanby scored the tying goal for Atlanta with 8:09 left in regulation, but the Sabres knocked off one of their top challengers in the East.
The Thrashers, first in the Southeast Division, dropped 10 points behind Buffalo in the conference standings with their third straight loss. Atlanta went 2-3-1 during a homestand at Philips Arena and now heads off on a six-game road trip, its longest this season.
Miller made 27 saves, including a brilliant stop in the final seconds of overtime. Kozlov swept in on a breakaway and picked out the top right corner, but Miller got enough of his glove on the shot to send it over the crossbar.
Atlanta dropped to 6-3 in shootouts, ending a streak of five straight wins. Buffalo improved to 8-2 in shootouts.