Jacobs, pinch-hitting, scored Willie Bloomquist, who had doubled.
Brian Bannister (4-1) got the win, allowing two runs and seven hits in six innings, helping the Royals end a four-game losing streak.
Bannister and Billy Butler each had an RBI for Kansas City.
Joel Pineiro (5-4) took the loss, yielding three runs and eight hits over seven innings.
Pineiro and Yadier Molina drove in the St. Louis runs.