"Vargy hasn't pitched in about nine days," manager Ned Yost said. "He went to a lot of 3-2 counts early, but then he got really locked in after the third inning."
Vargas threw 86 pitches, 56 for strikes, in his second outing since coming off the disabled list.
"I felt good," he said. "I was able to locate early and really use my curveball a little bit more tonight.
"I think just going out there and not feeling fatigued in the sixth inning was good, and it will be good for the next time out. Hopefully with the rest I've gotten, I can do it for them again."
And Vargas benefited from a Royals offense that beat Kluber for the third time this season.
"Our guys came up big," Vargas said. "They went up against a really good pitcher tonight so for them to score four runs was really big."
Jeffrey Flanagan is a reporter for MLB.com. Follow him on Twitter at @FlannyMLB. This story was not subject to the approval of Major League Baseball or its clubs.