With Cardinals forced an extra inning when Pete Kozma singled with one out in the bottom of the ninth, went to third on David Descalso's pinch-hit single and scored on Jon Jay's sacrifice fly.
Fernando Salas came in to pitch and retired Ian Desmond for the second out. The Cards elected to intentionally walk Danny Espinoza to get to Suzuki, but the Nats catcher made them pay by blistering a pitch into the left-center field gap for a two-run double.
The victory reduced Washington's magic number to clinch the National League East to one.
Michael Morse cracked a grand slam and Drew Storen (3-1) earned the victory despite allowing the tying run in the ninth.
Kozma finished 3-for-4 with two RBI for St. Louis, whose magic number for an NL wild card spot remained at three.