Gibson (1-0), a first-round 2009 Twins draft choice, gave up eight hits in six innings but only two runs. The 25-year-old right-hander, who had Tommy John surgery in November 2011, didn't issue a walk and struck out five.
Trevor Plouffe bashed a two-run homer while Justin Morneau and Pedro Florimon each added an RBI hit in the initial frame as the Twins grabbed a quick lead and cruised to their second win in three games against the Royals.
Plouffe supplied a sacrifice fly to finish with three RBI for Minnesota.
Wade Davis (4-6) was roughed up for six runs on five hits, lasting just one inning for Kansas City.
Alcides Ecobar and Salvador Perez each drove in a run for the Royals.