With the Yankees clinging to a 4-3 lead, Damaso Marte retired the first two Kansas City batters in the eighth before Veras was brought in to face pinch-hitter Billy Butler.
Veras walked Butler and was lifted for Coke, who gave up a double to Brayan Pena that scored pinch-runner Tony Pena Jr. with the tying run.
Alberto Callaspo added an RBI single. Buck, who homered in the third and finished with three hits, doubled home Callaspo with the final run.
Joakim Soria earned his third save by striking out the side in the ninth. Juan Cruz (1-0) worked a perfect eighth for the win, helped by a diving catch in right field from Willie Bloomquist.
Chamberlain, who made 12 starts last season before going on the disabled list with rotator cuff tendinitis and then going back to the bullpen, held the Royals to three runs - two earned - and four hits in six innings. He struck out five, walked one and hit two batters with pitches.
The Yankees took a 4-3 lead with a three-run seventh. Nick Swisher and Hideki Matsui opened the inning with singles, followed by Xavier Nady's RBI double. Royals first baseman Mike Jacobs committed an error on Robinson Cano's bouncer, allowing another run to score. Nady came around with the go-ahead run when Melky Cabrera grounded into a double play.
Royals right-hander Gil Meche held New York to three hits through six innings before giving up three runs and three hits in the seventh.
The Royals took a 3-1 lead with two unearned runs in the fourth after Swisher's error at first base with two outs. Callaspo's grounder went between Swisher's legs, scoring David DeJesus, who led off the inning with a walk. Buck singled home Mark Teahen, who was hit by Chamberlain's pitch, for the other run.
Buck had a solo homer in the third.
Brett Gardner, who led off the game with a single and stole second, scored the first Yankees run on Meche's wild pitch.
Meche went seven innings, allowing four runs and six hits with six strikeouts and two walks.
Notes: Yankees 1B Mark Teixeira was held out of the lineup for the second consecutive day with tendinitis in his left wrist. ...Royals 3B Alex Gordon did not play because of a sore right hip. ...The Royals are hitting a big league-low .201 with 14 runs in six games.
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