Mexico had looked the brighter in the opening stages of the game, with Carlos Salcido hitting the bar in the eighth minute and Andres Guardado firing just wide shortly afterwards, but Argentina went ahead through Carlos Tevez in the 26th minute despite the striker appearing to be offside.
The decision seemed to affect the players, and Ricardo Osorio then played a poor backpass that allowed Gonzalo Higuain to make it 2-0 in the 33rd minute. Tevez hit his second in the 52nd minute to put the game beyond Mexico's reach before Javier Hernandez grabbed a consolation goal.
"In the first few minutes, we were superior, but then we conceded a goal which was offside and then there was an error in the defence," Aguirre said. "Two mistakes made a dramatic difference.
"We got disorientated by the first goal, we lacked the mental strength and concentration to overcome the error that wasn't our own, and then we committed the second one. Referees and linesmen can take split-second decisions and they can spoil everything and years of hard work."
Despite the disappointment at the result, Aguirre - who took charge of the side when their qualification for the tournament looked to be in doubt - believes Mexico can be challengers at the 2014 tournament in Brazil.
"We have a great generation of young players who must be put to the best possible use," he said. "I believe in four years they will be at the top."