After spotting the Royals an early 2-1 lead, the Rays came back in the third.
Alex Torres (2-0), the first of four Rays relievers, notched the victory with 1 2/3 scoreless frames after starter Alex Cobb was hit in the head by a line drive of the bat of Eric Hosmer in the fifth inning.
Cobb was taken off the field on a stretcher and later diagnosed with a mild concussion.
Guthrie (7-4) was touched for five runs (four earned) on eight hits over seven full frames.
Salvador Perez knocked a solo homer for the Royals.
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