Aguilar lashed in a superb volley in the 118th minute to spark euphoric scenes at the iconic venue in front of 93,723 fans yesterday.
Aguilar's goal sends Mexico to Russia as CONCACAF's representatives in 2017 and piles pressure on US coach Jurgen Klinsmann who has faced criticism for a string of poor recent results.
But the victory capped a remarkable temporary stint as Mexico manager Ricardo Ferretti, who took the job on a temporary, unpaid basis following the sacking Miguel Herrera in July.
"I think in the end we did just a little bit more than them and deserved to win."
A downcast Klinsmann meanwhile said he would soldier on despite this latest setback, which followed elimination in the Gold Cup semi-finals in July and a heavy defeat to Brazil last month.
"Defeat is always difficult to swallow," Klinsmann said.
"The guys showed a lot of character, tremendous effort, they gave everything they had," he added. "I told the guys 'Heads up, because you gave everything you had.'"
Oribe Peralta looked to have settled the game with a 96th-minute strike that put Mexico 2-1 up on the night, ramming home Aguilar's low cross.
But the United States snatched an equaliser out of nowhere in the 108th minute when Bobby Wood sprinted onto DeAndre Yedlin's through ball to slip a cool finish beneath Moises Munoz to make it 2-2.