The Twins fell 1 1/2 games behind the Chicago White Sox in the battle for the lead in the AL Central.
Sweeney opened the inning with a single and moved to second on a sacrifice from Daric Barton.
Jesse Crain relieved Breslow and walked pinch-hitter Emil Brown on a 3-2 pitch. Suzuki then hit for Rob Bowen and ended the game with his double to left-center.
The walk-off hit made a winner out of Jeff Devine (4-0), who worked around a walk in the top of the ninth.