Hochevar (5-7) stretched his scoreless streak to 16 2/3 innings, allowing only one runner to reach as far as third base.
The right-hander surrendered one walk while striking out eight for the Royals, who were coming off a three-game interleague sweep at the hands of the St. Louis Cardinals.
Yuniesky Betancourt drove in three runs while Eric Hosmer provided a solo homer and a run-scoring single for Kansas City.
Alex Cobb (3-4) absorbed a complete-game defeat, coughing up eight runs on 13 hits for Tampa Bay, which fell for the fourth time in five games.
Celebrity Families of 2014 [PHOTOS]