Revere's third hit of the contest, a run-scoring infield single in the eighth inning, gave Minnesota an 8-5 lead and proved to be the game-winner as the Twins took the opener of a three-game series.
Willingham clouted a two-run homer in the first while Justin Morneau supplied a two-run shot and Trevor Plouffe also went deep for Minnesota, which pounded out 12 hits.
Jamey Carroll added three hits and a pair of RBI for the Twins.
Rookie Cole De Vries (1-1) earned his first major-league win despite surrendering five runs (four earned) on six hits over five innings. The right-hander didn't issue a walk and struck out four.
Matt Capps recorded the game's final out to earn his 13th save.
Royals starter Will Smith (1-2) was rocked for seven runs on eight hits in 4 2/3-innings.
Mike Moustakas went 3-for-4 with a solo homer, three RBI and three runs scored for Kansas City.