They also got a break from plate umpire Ed Montague in the eighth inning.
With two on and two outs, Jason Giambi appeared to check his swing on Mahay's 3-2 pitch near his ankles. But Kansas City catcher John Buck jumped up and tagged Giambi, and Montague called the slugger out swinging.
Giambi argued vehemently and Yankees manager Joe Girardi waved his arms in disbelief from the dugout, yelling "Come on, Eddie!"
Soria pitched a one-hit ninth for his 13th save in 13 chances, handing Darrell Rasner (3-3) a hard-luck loss. With its first road victory since May 18 at Florida, Kansas City improved to 2-19 in its last 21 visits to Yankee Stadium - and 6-33 since the start of the 1998 season.
New York lost for only the second time in its past nine home games.
Davies (2-0) gave up one run and seven hits in 6 2-3 innings.
The right-hander, who served up Alex Rodriguez's 500th home run last August at Yankee Stadium in his first start for the Royals, was called up from Triple-A Omaha last Saturday and beat Cleveland ace C.C. Sabathia 4-2 to end Kansas City's 12-game losing streak.
Rasner worked in and out of trouble all night, giving up three leadoff doubles and five two-baggers in all. But the right-hander limited the damage, allowing nine hits in a career-high eight innings. He threw 118 pitches and finished strong, retiring his final eight batters.
Royals shortstop Mike Aviles, from nearby Middletown, pestered the Yankees. Making his second big league start after being called up from the minors May 29, he doubled to right in the third for his first career hit.
The ball was taken out of play as a keepsake, but Aviles wasn't through. He hit another leadoff double in the fifth and scored on DeJesus' one-out single.
Jose Guillen opened the sixth with a double, went to third on Miguel Olivo's single and scored when Mark Teahen grounded into a hard-hit double play.
Robinson Cano doubled for the second time in the seventh and scored on buddy Melky Cabrera's two-out single, cutting it to 2-1 and chasing Davies.
