Wood (3-2) yielded only one run on five hits while overcoming a pair of walks. The left-hander allowed two or fewer earned runs for the fifth time in six starts, lowering his ERA to 2.33.
Nate Schierholtz provided Chicago's offense with a two-run homer in the fourth inning.
Cardinals starter Lance Lynn (5-1) was dealt his first loss of the season despite allowing only two runs on four hits over seven innings. He struck out eight.
Allen Craig cracked a solo homer for St. Louis.