Mark Ellis chipped in with a two-run double in the seventh to add some insurance for the Dodgers, who entered the contest 1 1/2 games behind the first-place San Francisco Giants in the National League West.
Billingsley (5-9) broke a personal five-start losing streak, holding the Cardinals to one run on seven hits over six innings.
Kenley Jansen picked up his 19th save as Los Angeles won the first of a four-game series.
Cards starter Joe Kelly (1-3) absorbed the loss, surrendering three runs (two earned) on four hits over six frames.
Carlos Beltran supplied a two-run homer for St. Louis.