Kansas City held a 4-2 advantage after seven innings but the Rays tied it up with two runs (one unearned) in the eighth on Ben Zobrist's one-out triple.
The Royals, however, came right back in the next half-inning when Butler belted a 2-1 pitch from Burke Badenhop (1-2) just over the left-field fence for the decisive blow.
Mike Moustakas also homered and Yuniesky Betancourt added a two-run single for the Royals, who outscored Tampa Bay 21-6 in their three-game sweep at Kauffman Stadium.
Aaron Crow (1-1) got the win after putting out the Rays' eighth-inning rally and Jonathan Broxton notched his 19th save for Kansas City.
Zobrist finished 3-for-4 with three RBI for Tampa Bay, which lost for the sixth time in seven games.
2014: The Year in Music [PHOTOS]