Greinke, who won the 2009 AL Cy Young Award, is 7-3 in his past 12 starts after starting the season 1-8. Greinke (8-11) held the Indians to one run and five hits over six innings, lowering his ERA to 3.90.
Betancourt homered, his 11th, to left in the fifth to tie the score at 1. Betemit put the Royals ahead to stay by leading off the sixth with his sixth home run, which landed in the back of the Royals' bullpen.
Indians rookie right-hander Jeanmar Gomez (3-1) was handed his first loss in five big league starts. Gomez allowed seven hits in six innings.
Travis Hafner, who came off the disabled list Sunday, led off the Cleveland second with a double, stopped at third on Jayson Nix's single and scored on Matt LaPorta's groundout.
The Indians wasted several scoring opportunities, stranding 12 and going 1 for 9 with runners in scoring position.
Jesse Chavez, who replaced Greinke, walked the bases loaded with two outs in the seventh before striking out Nix to end the inning.
Hafner led off the sixth with a walk and stopped at third on Trevor Crowe's one-out double. Greinke retired LaPorta on a grounder to third and Jason Donald on a fly to right to strand the runners.
Joakim Soria worked around a leadoff single by Michael Brantley in the ninth to earn his 34th save, which tied for the AL lead, in 36 opportunities. Soria has saves in 27 consecutive chances since a blown save May 6 at Texas.
NOTES: Royals RHP Brian Bannister, who has not pitched since Aug. 2, was placed on the 15-day disabled list with right rotator cuff tendinitis. Bannister is 7-11 with a 5.95 ERA in 22 starts. Kansas City recalled OF Jai Miller from Triple-A Omaha, where he was hitting .267 in 84 games. Miller batted .345 with six home runs and 15 RBIs in 15 August games in the Pacific Coast League. ... 3B Josh Fields, out all season following hip surgery, had his minor league rehab assignment transferred by the Royals to Double-A Northwest Arkansas after going 4 for 10 with four RBIs in three games with Surprise in the rookie Arizona League. Cleveland's Shelley Duncan, in an 0-for-19 skid, grounded out as a pinch hitter to end the eighth. The Indians and Royals met for the first time since May 20. There were seven new faces in the lineups Tuesday that did not play in that game May 20.