Goals from Landon Donovan and Brian McBride saw the USA progress to the quarter-finals at the expense of their rivals.
Mexico, who had impressed in reaching the knockout phase, were unable to puncture a resolute USA backline.
We tried everything but came up against a brick wall
Mexico striker Jared Borgetti
But Aguirre believes his side were denied a clear penalty after American John O'Brien appeared to handle the ball in the area in the second half.
"We were unlucky, we conceded the goals at the worst possible time for us," said Aguirre.
"Then just when we were getting back on top of the game there was a very unfortunate decision by the referee.
"They showed the replay on the big screen and we saw it, 40,000 fans saw it, but it's over now." Aguirre also labelled Bruce Arena's USA side defensive.
"They didn't want to play, they didn't let us play," he added.
"Croatia and Italy came out to play. They [the USA] set out not to lose."
Despite going into the game as favourites to make it through to the last eight, Aguirre dismissed suggestions they had underachieved.
"We came here to improve on our 13th place in France and the 13th place we obtained in the USA," he said.
"I don't know how we did but I think we fulfilled a promise - no more, no less."
Mexico striker Jared Borgetti congratulated the Americans.
"They scored with their first chance and played well," he said.
"We knew they were a team that counter-attacks well and they have fast players.
"I think that we played good football but it is necessary to score. It is possible to destroy good football by playing defensively.
"We tried everything but came up against a brick wall. We kept insisting, looking for a goal, but they held on."
Rats and idiots
Back in Mexico millions of fans stayed up through the night to watch the game.
And defeat was hard to take against a country they view with much ill-feeling.
"It hurts us here," said Mexico fan Jose Luis Luviano.
"There has to be an end to this disgrace where they (Americans) treat us like rats and idiots."
President Vicente Fox summoned almost his entire cabinet to watch the match, which kicked-off at 1.30am local time.
And Fox remained defiant after the loss.
"In no way do we feel defeated," he said. "The important thing is to fight, fight, fight with tenacity."