The struggling Royals have lost five in a row and seven of the last eight. Tampa Bay has won four of five.
The Rays jumped on Kansas City starter Jason Hammel quickly, scoring three times in the first inning. Tampa Bay loaded the bases with one out. Denard Span scored on an RBI groundout by Brad Miller, and Hechavarria drove in Cron and Wilson Ramos with a two-run single.
Cron extended the Rays' lead to 4-0 in the third inning, with his 10th home run of the season, a solo shot to left field off Hammel.
Kansas City got on the board in the fourth inning. Alcides Escobar's sacrifice fly brought home Whit Merrifield.
The Rays got the run right back in the fifth inning on Ramos' RBI ground-out that scored Cron.
The Royals rallied for two runs in the bottom of the fifth inning, knocking Rays' starter Jacob Faria out of the game. Jon Jay opened the inning with a double and came home on Jorge Soler's RBI double.
Soler then stole third, putting himself position to score on Mike Moustakas' sacrifice fly and cutting Kansas City's lead to 5-3 heading into the sixth inning.
Faria scattered three runs on four hits, with two strikeouts and four walks in 4 1/3 innings.
Hammel surrendered five runs on 10 hits, with two strikeouts in six innings. He did not walk a batter.
The Rays head to Anaheim to put a wrap on its 11-game road trip against the Los Angeles Angels. The four-game series begins Thursday. Chris Archer is expected to take the mound for Tampa Bay.
The Royals welcome the first-place New York Yankees to Kansas City for three games, beginning Friday. Jakob Junis will get the start in the series opener against the Yankees.
--Field Level Media