The Royals won at home for the first time this season after losing their first 10 in Kansas City.
The night started and ended poorly for the Yankees, who lost superstar reliever Mariano Rivera prior to the contest -- for the evening and perhaps for more than that.
Rivera injured his right knee while catching fly balls in the outfield during batting practice. He was taken to a hospital for further examination.
Kansas City never trailed, getting three RBI from Mike Moustakas on a solo homer in the second and a two-run single in the fifth.
Mark Teixeira drove in two runs for the Yankees, the second of them in the seventh inning on a sacrifice fly that cut New York's deficit to one run.
Broxton came on in the ninth as Kansas City's fifth pitcher and began the inning by giving up a single to Derek Jeter and walking Curtis Granderson. He got Teixeira to ground into a double play and Rodriguez followed with a groundout to end the game.
It was the fifth save for Broxton and made a winner out of starter Danny Duffy (2-2), who allowed two runs on six hits in 5 1/3 innings.
The loss went to David Phelps (0-1), who surrendered two runs on six hits in four frames. Phelps made his first start of the season after replacing Freddy Garcia in the rotation.