Kazmir (8-8) lasted only four innings, giving up four runs (three earned) while striking out four and walking none.
Shields (11-9) bounced back from a bad start Sept. 6 against Detroit — where he surrendered 10 runs — to stifle the Indians' offense. He held the Indians to just two runs on four hits, striking out seven and walking one in eight innings.
Shields improved to 2-5 in 13 career starts against Cleveland. He had no-decisions in his first three starts against the Indians this season, allowing nine earned runs in 16 1-3 innings.
After the first inning, Shields pitched like the ace the Royals envisioned he would be when they acquired him from the Rays for outfielder Wil Myers, one of baseball's top prospects. The right-hander leads the AL with 207 2-3 innings pitched.
Shields' biggest problem came with his own defense, when he made two errors in the third. He dropped a throw at first base, allowing Kipnis to reach safely. He then threw wildly to first trying to pick off Kipnis, but retired Carlos Santana to end the inning.
The Indians would answer back with both of their runs in the bottom of the first. Bourn led off with a single and advanced to second after Mike Aviles was hit by a pitch. Kipnis' groundout moved both runners up, Carlos Santana popped out to third and Michael Brantley delivered with a two-out, two-RBI single.
After that, the Indians went 0 for 23 with a walk before Kipnis led off the ninth with a single against Shields. Santana followed with a blooping single to left, chasing Shields.
Closer Greg Holland came on and struck out the for his 42nd save.
"They are a good offensive team. But we're facing a lot of teams now that are stacked," Kazmir said. "That's no excuse. We still need to get the job done."
NOTES: Indians 1B Nick Swisher wasn't in the lineup against Shields. He is batting .167 (8 for 48) with 15 strikeouts against the right-hander, including three hits in his last 25 at-bats. ... SS Asdrubal Cabrera, in a 3-for-27 slump, was also out of the lineup. He pinch-hit in the ninth and struck out. ... Indians RHP Corey Kluber (8-5) faces LHP John Danks (4-12) in the series opener against the White Sox.