Altidore raced clear down the right and crossed for Dempsey, whose point-blank effort was parried by the goalkeeper.
The U.S. went ahead when Jozy Altidore played the ball to Jones, who flicked the ball over the defense.
Fed by Altidore, his first effort crashed back off the post, and his second was slashed wildly wide with left foot.
Stephen Hunt powered in a shot for the home side after Jozy Altidore's effort had been parried by keeper Tim Howard.
Dempsey went close again before Jozy Altidore missed the clearest opening of the half, blazing wastefully over the bar from five yards.
Bullard, making his full debut, went close with a fizzing 30-yard drive and Jozy Altidore dragged wide from the edge of the box.
Hull completed their first summer signings on Thursday by bringing in midfielder Seyi Olofinjana from Stoke and Villarreal striker Jozy Altidore on loan.
"Between getting batteries and bags of urine thrown at me, I think I've probably seen it all, " striker Jozy Altidore said before the game.
Klinsmann even dropped Altidore from a pair of qualifiers last fall.
England, however, were not looking comfortable and Altidore illustrated the lack of pace at the back to give them a real scare in the 64th minute.
The United States were allowed the luxury of plenty of possession, and England almost paid for their generosity when Landon Donovan's inviting cross was headed tamely wide by Altidore.
Hull went ahead after Everton defenders Sylvain Distin and Leighton Baines missed a deflected Hunt cross to allow Altidore to hammer a point-blank shot, which Howard did well to fend away.
Altidore headed wide from a Garcia centre and then, having been played in by Bullard, opted to square the ball to nobody instead of shooting from a central position 12 yards out.
Midfielder Michael Bradley, giving his usual fearless and indefatigable effort, delivered a deft pass on the left side to Fabian Johnson, one of Klinsmann's German transplants, who sent the ball into the middle of the penalty area to Jozy Altidore.
It exposed a glaring lack of pace in England's central defence, which was almost punished when Jozy Altidore raced past Jamie Carragher with embarrassing ease only for Green to apply some repairs to his shattered reputation by turning his shot on to the post.
Playing with a newfound calm and touch, Altidore collected the ball, held it just long enough for Dempsey to break free, then laid off a perfect ball that Dempsey slammed into the lower left corner for a goal that left the Azzurri dumbfounded on home soil.
Several players have been targets of abuse from supporters in recent weeks, including American forward Jozy Altidore of the Dutch club AZ Alkmaar, Barcelona defender Daniel Alves and AC Milan midfielder Kevin-Prince Boateng, who led his teammates off the field during an exhibition game in protest.
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