Cain's triple in the second inning put the Royals ahead 3-0 and Escobar's eighth-inning three-bagger highlighted a four-run outburst that powered Kansas City to a victory in the opener of a three-game set.
Eric Hosmer went 3-for-5 with a homer and three runs scored while Salvador Perez added two hits and a sacrifice for the Royals.
Smith (5-7) scattered seven hits, walking three and striking out seven en route to the win.
Twins starter Scott Diamond (11-7) was tagged for 10 hits over six full frames, allowing four runs.
Ryan Doumit drove in Minnesota's only run.