The Indians have outscored the Royals 15-1 in the first two games of this series. The loss dropped the Royals back to .500 and 7 1/2 games behind the first-place Indians.
Bauer (12-8) gave up seven hits and two walks in 6 1/3 innings with four strikeouts. Shaw and Allen did not allow a baserunner.
Jason Kipnis and Jose Ramirez hit back-to-back doubles in the first inning to produce a Cleveland run. Ramirez, who did not play Friday after being hit in the right forearm pitch Thursday, drilled a Jason Vargas pitch to center, scoring Kipnis.
The Indians loaded the bases in the fourth with one out on a walk and singles by Santana and Jackson. It appeared Vargas would wriggle out of this jam when he struck out Brandon Guyer swinging on a full-count pitch for the second out.
The Indians tacked on a run in the fifth and chased Vargas.
Ramirez walked with one out, advanced to second on Edwin Encarnacion's groundout and scored on Bruce's double to right on Vargas' 96th and final pitch.
Vargas, who has only two wins since June 30 after being named to the All-Star Game, yielded four runs, six hits and three walks while striking out seven in 4 2/3 innings. He was replaced by Mike Minor.
The Royals had opportunities early but came away with zilch. Lorenzo Cain doubled with one out in the first and Brandon Moss with two outs in the fourth but went no farther. Drew Butera and Cain singled in the fifth, but Bauer prevented them from scoring.
NOTES: The Royals placed RHP Neftali Feliz on the disabled list with right ulnar nerve palsy. LHP Eric Skoglund was recalled from Triple-A Omaha, where he was 4-5 with a 4.07 ERA in 20 starts. Skoglund will be used as a reliever. ... Indians manager Terry Francona said according to the training staff it is 50-50 whether RHP Corey Kluber will make his Wednesday start against Boston. Kluber exited in the sixth inning on Friday with a sprained right ankle. ... Royals C Salvador Perez took a full batting practice before the game for the first time since going on the DL with a right intercostal strain. ... Royals manager Ned Yost celebrated his 62nd birthday. ... RHPs Danny Salazar of the Indians and Jason Hammel of the Royals are the Sunday probables.