Vargas (7-3), who missed most of last season after having Tommy John surgery in 2015, last had a complete-game shutout on Aug. 13, 2014, against Oakland. He limited the Indians to seven hits, six of them singles, and was backed by four double plays.
Johnny Cueto threw the last complete-game shutout for the Royals on Aug. 10, 2015, against Detroit.
Lorenzo Cain beat out an infield tapper to the left side that Indians pitcher Josh Tomlin fielded and threw to first in the sixth inning. Umpire Ed Hickox ruled Cain out, but the Royals challenged and the call on the field was overturned after a 75-second delay. The Royals have won 11 of 14 challenges this season.
Cain's infield single with two outs scored Alcides Escobar with the game's first run.
Escobar had singled with one out, took second on a Tomlin wild pitch and advanced to third on Mike Moustakas' groundout before scoring on Cain's dribbler.
The Royals added two runs in the eighth on a home run by Mike Moustakas and an RBI double by Eric Hosmer.
Tomlin (3-7) gave up three runs and seven hits in 7 1/3 innings.
Tomlin, who threw a complete-game six-hitter to beat the Royals on Sunday in Cleveland, breezed through the first five innings, holding Kansas City to two singles and no runs.
Jorge Bonifacio and Brandon Moss singled with two outs in the second for the only time Tomlin was in any kind of danger the first five innings. Whit Merrifield flied out to right fielder Daniel Robertson to end the inning.
Vargas matched Tomlin on putting up zeroes. The Indians managed six hits and a walk off Vargas the first six innings but failed to score.
Cleveland grounded into four double plays in the first six innings. Carlos Santana grounded into double plays in the first and fourth innings.
Yan Gomes' double off the left-field fence with one out in the Indians' fifth was the lone extra-base hit. Vargas struck out Austin Jackson and retired Robertson on a grounder to Alcides Escobar to strand Gomes.
NOTES: Indians OF Lonnie Chisenhall (concussion) could go to Double-A Akron for a rehab assignment this weekend if he gets clearance by the doctors. ... The Royals finished May with a 15-14 record. "Better than April, it's a start," Royals manager Ned Yost said. "We need to find a way to get on a nice run where you win 15 out of 22." ... Royals OF Jorge Soler has started only one game since May 23. ... Indians RHP Carlos Carrasco and Royals RHP Jason Hammel are the Saturday probables.