Cain's RBI groundout three batters in put the Royals ahead, and Kansas City expanded its lead to 3-0 in the sixth when Alex Gordon hit an RBI double off Hudson and scored on Eric Hosmer's single off left-hander Javier Lopez.
San Francisco finally got to Guthrie in the bottom of the sixth when pinch-hitter Michael Morse hit an RBI double over third base just past the glove of a diving Mike Moustakas and scored on Buster Posey's groundout against Kelvin Herrera.
Herrera, Brandon Finnegan, Wade Davis and Greg Holland combined for four innings of hitless relief, the longest in a Series game in 22 years.