Mike Aviles added a solo home, and David DeJesus hit two doubles and drove in a run for the Royals.
Carl Crawford singled in the Tampa Bay run.
Kyle Davies (4-2) got the win. Davies allowed a run and six hits before Kansas City relievers took over and tossed four scoreless innings.
The Royals got five runs and eight hits off Tampa Bay starter and loser Andy Sonnanstine (10-6), who lasted 6 2/3 innings.