Kansas City, which lost a major league-high 106 games last year, hasn't won at Yankee Stadium since August 2002. Jason Giambi hit an early three-run homer to help extend the Royals' losing streak in the Bronx to 12.
With the Royals ahead, 7-4, Giambi fell behind 1-2 in the count against Andrew Sisco leading off the eighth, then walked. Hideki Matsui singled and Jorge Posada got the Yankees' ninth walk of the game, loading the bases.
Robinson Cano hit an RBI groundout, beating the relay to first to avoid a double play. That brought up Williams, who received more standing ovations from Yankees fans who worried last fall that he wouldn't be re-signed.
He cost New York in the fourth when he was doubled up at second on a popup, but this time he punched a run-scoring single to left.
Ambiorix Burgos relieved to face Damon, playing his first home game in pinstripes after spending four seasons with the Red Sox. Damon slammed his bat in the dirt after swinging under a belt-high fastball, and Jeter came to the plate.
''It seems when something needs to happen, he seems to be at the start of it or the end of it," Yankees manager Joe Torre said.
Yet in previous home openers, Jeter had no homers and two RBIs in 34 at-bats. The Yankees' captain was looking for a fastball, but jumped on a splitter. ''They're all special," said Jeter, who refused to rank the great moments he's had in this ballpark. ''It seems like the season hasn't really officially started until we play our home opener."
Scott Proctor (1-1) pitched one inning for the win, and Mariano Rivera finished for his first save.
Rivera had no doubt Jeter would come through.
''I've seen it for years," Rivera said. ''That doesn't surprise me at all."
