Snapping their three-game winning streak, the Yankees fell five games back of the Red Sox, who defeated the Devil Rays on Monday behind starter Tim Wakefield. New York also dropped 1 1/2 games behind Seattle, which beat Minnesota on Monday, in the American League Wild Card race.
The Yankees came up on the losing end behind their fifth pitcher, Henn, who recorded the first out of the 10th but surrendered a double to Howie Kendrick.
Budde, who entered the game as a defensive replacement after the Angels pinch-hit for catcher Jeff Mathis, dropped the decisive hit between the lunges of Melky Cabrera and Bobby Abreu in right-center field, sparking an on-field celebration that kicked off the first game of a three-game series for the Yankees at Anaheim.
"He hit it and it fell," said Henn (2-1), who described watching the play as being in slow motion. "That's the way it works."
The abrupt ending made a winner of left-hander Darren Oliver (1-0), who pitched a scoreless 10th, and nullified great offensive performances from Alex Rodriguez and Jorge Posada, each of whom homered and drove in three runs for New York.
After starter Dustin Moseley limited the Yankees to two runs over five innings, Rodriguez hit his Major League-leading 40th home run off Chris Bootcheck in the sixth. With the Yankees trailing, 6-4, in the eighth, Posada came through with his 16th home run, a two-run shot to right off Justin Speier.
Local product Phil Hughes, a 2004 first-round pick from Foothill High School in Santa Ana, Calif., started for New York. Leaving a total of 33 passes for family and friends, Hughes held the Angels to just Mathis' three-run double through his first six innings.
"It was cool," Hughes said of pitching at Angel Stadium. "I've seen more games at this stadium than any other one, so it felt more familiar."
Pitching, as Torre said, his most competitively since returning from injuries that kept him out of Major League action for the better part of two months, Hughes saw his homecoming run into trouble in the seventh.
Kendrick opened the inning with a jam-shot single and moved to second when Hughes issued his career-high fifth walk of the night, prompting Torre to call upon Luis Vizcaino for a two-on, one-out situation.
"It was tough," Hughes said. "The walks will kill you. Fortunately, I was able to keep my pitch count down to give us a chance late in the game. You look back, and they had maybe two hard base hits and five runs on the board. It's hard to let it happen, but sometimes it just does."
As he walked off the field, Hughes was visibly upset, a rare display of emotion from a hurler who has proven normally even-tempered.
"There's no question he's special," Torre said. "Whether he's pitching here, at [Triple-A] Scranton or [Double-A] Trenton, wherever, he feels he's capable of doing certain things."
Chone Figgins greeted Vizcaino with a line-drive single to center field on a pitch that caught too much of the plate, scoring Kendrick from second base. Orlando Cabrera followed by punching a hot single to right, scoring Reggie Willits, and Vladimir Guerrero drove in the Angels' sixth run with an infield groundout.
With Kyle Farnsworth back in the eighth-inning role that he eventually lost with poor results, the Yankees nearly watched the tie zip away in the eighth. Farnsworth was saved by first baseman Wilson Betemit, who stabbed a Maicer Izturis grounder and cut down Gary Matthews Jr. at the plate.
"He was so cool making that play," Torre said of Betemit, who entered the game after Jason Giambi had pinch-hit for starter Andy Phillips. "That's the infielder in him."
Farnsworth then struck out Willits on a disputed check-swing to end the eighth, a play on which Scioscia was ejected by third-base umpire Dan Iassogna.
Mariano Rivera pitched around trouble in the ninth inning for New York, but the Yankees could only hold off the pesky Angels that long.
"It was just one of those games someone was going to have to win," Torre said. "It's tough to take, but both clubs can't win. It's tough to lose games like that, but you understand it's going to happen."