Garcia (7-7) tamed the National League Central champs on a pair of runs, issuing only one walk and striking out six in a clutch performance for the Cards.
He also smacked a game-tying solo homer in the third inning, triggering a three-run uprising that handed control of the contest to St. Louis.
Edward Mujica recorded the final out of the seventh, Mitchell Boggs tossed a scoreless eighth and Jason Motte secured the victory with his 41st save as the Cards claimed the opener of a season-ending three-game series.
With the Los Angeles Dodgers' win over San Francisco Monday, St. Louis needs one more victory or one loss by the Dodgers to clinch the second National League wild card berth.
Cincinnati's Bronson Arroyo (12-10) was reached for three runs on six hits over five full frames, absorbing the loss.