The Cardinals lost for the fifth time in six games and slipped out of a first-place tie with Pittsburgh in the National League Central.
Oakland sent 10 men to the plate in the second, battering Shelby Miller (8-6) en route to a third straight win.
Josh Donaldson singled home the first run of the inning and then scored on a double by Josh Reddick. After Eric Sogard and Coco Crisp drew walks to load the bases, Smith delivered his clutch single to center that gave the A's a 4-0 lead.
Jed Lowrie drove Miller from the game with a double that brought home Crisp.
Bartolo Colon (11-2) worked eight innings to get the win, allowing a run on six hits. Colon has won his last eight starts, having suffered his two losses this season in consecutive appearances on May 4 and May 9.