Royals manager Ned Yost, after Kansas City's 5-4 win over Tampa Bay on Wednesday night, said the surgery went great and that there were "no complications."
Yost said Vargas "started really having severe discomfort" around 4 a.m. Wednesday.
There was no timetable given for how long Vargas will be out. Bruce Chen could replace Vargas in Sunday's game against Detroit.
Royals All-Star outfielder Alex Gordon was scratched from Wednesday's lineup due to a sprained right wrist. He will have an MRI and see a hand specialist Thursday when the team returns to Kansas City.
Gordon did enter as a defensive replacement Wednesday in the ninth inning. He said the major problem is pain when swinging a bat.
"I don't think it's anything serious," Gordon said.
Yost said the injury could "very well" affect Gordon's status for next week's All-Star Game.
Kansas City third baseman Mike Moustakas also was out of Wednesday's lineup with the flu. He was feeling better late Wednesday but is not expected to start Thursday.