Alcides Escobar had three hits and Omar Infante also drove in two runs for the Royals, who followed up a sweep at the hands of Minnesota last weekend by sweeping Houston and then taking the first of their three-game set against the Twins this weekend.
Ricky Nolasco (1-2) gave up five runs on 11 hits in 5 2-3 innings for the Twins. He was coming off a 7-1 win over Kansas City in which he allowed one run on four hits in eight innings.
Vargas (2-0), who signed a $32 million, four-year deal in the offseason, allowed seven hits while striking out four in another dazzling start. The crafty left-hander lowered his ERA to 1.24 while going at least seven innings in each of his four outings this season.
Vargas put runners on base each of the first five innings, though he was never in serious trouble. Only twice did a runner reach second base and each time a lazy fly ball ended the inning.
Even when the Twins hit a ball hard, Alex Gordon was there to make a play.
The Royals' Gold Glove left-fielder threw a strike from the warning track to second base in the fourth inning, cutting down Josmil Pinto as he tried to stretch a single. Two innings later, Gordon made a running catch into the padding in foul territory on Joe Mauer's fly ball.
Escobar sent a charge through the Royals' anemic offense with a double in the third inning, and consecutive singles by Nori Aoki, Infante and Eric Hosmer staked them to a 2-0 lead.
Infante's single drove in Escobar to make it 3-0 in the fifth.
Billy Butler, in the throes of a massive slump, singled in the sixth before Moustakas, also off to a slow start, sent a pitch sizzling into the seats in right. His second homer of the season made it 5-0 and chased Nolasco from the game.
Neither team threatened to score again, the Royals wrapping up their first shutout win and the Twins getting blanked for the first time this season.
NOTES: The Royals placed OF Jarrod Dyson on the bereavement list and recalled INF/OF Jimmy Paredes was recalled from Triple-A Omaha before the game. ... An MRI exam on Royals OF Lorenzo Cain (strained groin) came back negative. Manager Ned Yost hopes to have him back after a 15-day DL stint. ... Twins OF Jason Kubel got the night off from the starting lineup. He flied out as a pinch hitter in the seventh. ... LHP Bruce Chen (sore back) will start for KC on Saturday. He was initially scheduled to pitch Thursday. RHP Kevin Correia will start for the Twins. ... There was a moment of silence before the game for the victims of the shootings Sunday at a Jewish community center and nearby senior living center in suburban Kansas City.