Vargas (1-0) allowed just one run -- a solo homer in the ninth -- among four hits to earn the win. The southpaw has allowed just two runs this season.
Greg Holland relieved Vargas in the ninth and allowed one run, but settled down to earn his third save of the season.
Matt Moore (0-2) left the game in the fifth inning with an apparent injury. The southpaw delivered a pitch to Royals outfielder Norichika Aoki and grimaced in pain. He was visited at the mound by manager Joe Maddon and the Rays training staff before exiting.
He allowed one run on four hits before leaving.