"I feel calm now, the pressure of the last few days has gone and I think I will go and have a couple of shots of tequila," he told reporters after substitute Miguel Sabah's 83rd-minute goal sealed victory for the hosts.
The result left the 105,000 home crowd at the Azteca stadium in a party mood, after frustration during large stretches of the match with Mexico unable to turn their domination of possession into a winning goal.
The U.S. went ahead through Charlie Davies in the ninth minute but Israel Castro struck back 11 minutes later.
"The only real mistake we made was for the American goal -- that can happen because you can't be perfect for 90 minutes," said Aguirre.
The coach said the days of dressing room cliques, fierce media criticism and fan unrest during the early stages of Mexico's qualification campaign under predecessor Sven-Goran Eriksson were over.
"I am extremely happy with the win. This is a victory which has brought together everyone - the team, the fans and the media. Everyone did their job for us out there and they can sleep well tonight," he said.
Mexico began the day in fourth place in the CONCACAF qualifying group and with only the top three heading automatically to next year's finals in South Africa, the three points could prove invaluable.
"There are still 12 points to play for, it's difficult to say that we have the ticket to South Africa in our hands but we did what we had to do today," Aguirre said.
"There is a lot of work remaining, the team has to defend better and to take their chances in front of goal.
"There is no way that you can play beautiful football in qualification, you just have to win, forget about beautiful play and just get the results."
Mexico's next opponent is group leader Costa Rica on Sept. 5, a game Aguirre forecast as "tricky".
"We can allow ourselves a couple of shots tonight but the work for that game starts tomorrow," he said.
