"I felt like I had good at-bats all night but had nothing to show for it," Rhymes said. "I just kept grinding and it felt really good to get a big hit right there. We didn't need to play any more."
The Tigers, who have played three straight extra-inning games, won 10-9 in 13 innings Thursday at Minnesota and did not arrive in Kansas City until around 4 a.m. Friday.
"It's been a long couple of days," Rhymes said. "Fortunately, we found a way to come out on top."
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.
"The error didn't help, but Jesse is making too many pitches up and over the middle of the plate and good hitters hit those pitches hard," Royals manager Ned Yost said. "He needs to find a way to get back down in the zone on the outer third to be effective."
Ryan Perry (3-5) picked up the victory with two perfect innings.
"I came out with a chip on my shoulder," said Perry, who took the loss in the 10th inning Wednesday. "I threw with a lot more aggression, but under control. It definitely feels a lot better than walking off with a loss."
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.
"Greinke's one of the best in the business and he had good stuff," Tigers manager Jim Leyland said. "Coming into the fourth inning we were really starting to worry about a no-hitter. He had that kind of stuff."
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 whole game was nothing special and they started hitting the ball a little harder in the seventh inning," Greinke said. "I didn't really pitch great the whole game, so it just kind of evened out the last inning."
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.
"You got take off your hat to Bonderman, his perseverance and staying with it after that four-run second inning and you look up in the eighth and he's still pitching," Royals manager Ned Yost said.
Said Bonderman, "I was fighting myself pretty much all the way into the third. I wanted to finish strong. It was like a totally different game after that. I thought I was in trouble with Zack Greinke on the mound and behind 4-0. But we found a way to win."
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.