Guthrie (7-3) turned in a third straight quality start, yielding two runs on six hits and three walks, exiting with a one-run lead.
The Royals relief corps made the advantage hold up as Aaron Crow, Tim Collins and Kelvin Herrera combined for 1 2/3 scoreless frames and Greg Holland tossed a 1-2-3 ninth for his 13th save.
Salvador Perez delivered a two-run triple in the third inning and scored on Lorenzo Cain's single as the Royals took the opener of a three-game series.
Detroit's Doug Fister (5-4) was tagged with a complete-game loss, surrendering three runs on nine hits. He didn't allow a walk and struck out three.
Miguel Cabrera belted his 18th homer of the season -- a two-run shot -- for the American League Central-leading Tigers.