Rhymes' triple to right-center field off Jesse Chavez (2-3), the sixth Royals pitcher, scored Alex Avila, who had singled, and Casper Wells, who reached on a Josh Fields error. After Johnny Damon walked, Ryan Raburn singled to right to score Rhymes. Austin Jackson's suicide squeeze bunt scored Damon with the final run.
Ryan Perry (3-5) picked up the victory with two perfect innings.
Raburn, who has homered in back-to-back games, homered on the first pitch from Royals rookie reliever Blake Wood leading off the eighth, putting the Tigers ahead 5-4.
Brandon Inge hit a two-run single as part of the Tigers' four-run seventh. Avila followed with a single to score Kelly, who had three hits. Avila scored on Robinson Tejeda's wild pitch to tie the game 4-4.
The Royals put up their own four-run inning in the second. Brayan Pena singled home Fields, who led off the inning with a single, before Yuniesky Betancourt hit a two-run single to push his RBI total to 67. Gregor Blanco followed with a triple to score Betancourt.
Tigers right-hander Jeremy Bonderman, who missed his previous start with inflammation in his right rib cage, allowed four runs on seven hits and a season-high five walks in 7 1-3 innings.
Notes: C Gerald Laird was a late Tigers scratch with back spasms. ... 1B Miguel Cabrera was held out of the Detroit lineup because of left biceps tendinitis. ... Fields, who had hip surgery in April, made his first appearance of the season and singled in his first at-bat. ... Royals C Jason Kendall is expected to be out eight to 12 months after having right shoulder surgery Friday in Los Angeles to repair extensive tearing of the rotator cuff. ... RHP Brian Bannister, who was reinstated from the Royals disabled list, will start Wednesday at Minnesota, replacing RHP Bryan Bullington in the rotation. Bannister was out with right rotator cuff tendinitis.