Guthrie (4-6) allowed just one run on four hits and a walk while tying a season-high with nine strikeouts to win back-to-back outings for the first time since starting the season 2-0.
Omar Infante connected on a solo home run and Alex Gordon drove in a run for the Royals, who have their first 10-game winning streak since a 14-game run July 23-Aug. 5, 1994. They will go for the four-game sweep Thursday at Comerica Park.
Greg Holland worked around a leadoff base hit to secure the win and capture his AL-leading 21st save of the season.
Drew Smyly (3-6) was the tough-luck loser after tossing seven-plus innings and yielding two runs on seven hits while fanning six.
J.D. Martinez's solo shot was the lone run for the Tigers, who have dropped three straight and 12 of their last 17 contests.