The Royals have won four straight and six of seven to move three-games above .500 for the first time since May 12, 2011, when they were 20-17.
Alcides Escobar's single scored Chris Getz. Salvador Perez had three of the nine Kansas City hits.
Davis (1-0) held the Twins to four singles in five scoreless innings, striking out six, walking three and hitting a batter.
Twins right-hander Liam Hendriks (0-1) took the loss.