Kluber improved to 4-1 against the Royals in his career. He gave up just four hits and struck out 10 in 8 1/3 innings. That gave Kluber 136 strikeouts, fourth most in the American League.
"He's having an outstanding year and he was better than I was today," said Royals starter Danny Duffy.
The Indians struck at Duffy with two home runs in the second inning. Carlos Santana opened the inning with a drive into the right-field bullpen, his 13th homer this season.
Ryan Rayburn lined a single to left field and, after an out, Yan Gomes sent a blast soaring over the center-field wall for two more runs. It was No. 10 for Gomes.
Initially, Duffy didn't think either ball would leave the park.
"I didn't execute on the changeup to Santana. I didn't think it was out off the bat but, sure enough, it was," Duffy said. "And then, Gomes hit a pitch that was a little elevated, belt-high, and I didn't think that ball was going out either. … I didn't think he really got it, but any ball that leaves the park is got."
Duffy got help in the third inning from third baseman Mike Moustakas. Michael Brantley was on first with an infield single when Santana bounced a rocket that Moustakas snagged and started a lightning-fast double play to end the inning. He made another dazzling stop on Gomes to close the eighth inning for Bruce Chen.
"He was really flashing the leather today. He always does," Duffy said. "Moose is a savage at third base."
Moustakas also accounted for the Royals' only run with a home run, just inside the right-field pole, in the fifth inning. It was his 10th this year. He also worked Kluber for his only walk on a 10-pitch at-bat.
"I was able to get a pitch, one of the few mistakes I think he made on the day and I just didn't miss it," Moustakas said. "On the walk, he was throwing some nasty stuff up there and I was just trying to stay alive."
Moustakas is experiencing something of a revival at the plate.
In his last 21 games, he's 20-for-69, .290, with six homers and 15 RBIs.
"Defensively, he was spectacular," manager Ned Yost said. "Offensively, he put the only run on the board for us. He's really been swinging the bat much better. That's why we hung with him through the rough time offensively -- because he's such a good defender at third."
Duffy went six innings, giving up four runs on 10 hits with no walks and six strikeouts.
"This was a good outing for Danny," Yost said. "He kept us in the game. A four-run outing by your starter is not insurmountable. He did a real fine job."
Newcomer Scott Downs pitched a perfect seventh inning against the top of the Indians' order. Chen added a scoreless eighth.
Kluber went for a complete game -- the only one of his career came on April 24 against the Royals. But after Eric Hosmer doubled to start the ninth and Salvador Perez became Kluber's 10th strikeout victim, manager Terry Francona pulled his pitcher to a chorus of boos from Indians fans.
"I don't blame them. If I was managing with my heart, I'd have left him in there, too," Francona said.
His departure was fine with the Royals, though. They'd seen plenty of Mr. Kluber.