The 50-year-old had replaced ex-England manager Sven-Goran Eriksson in April 2009 and resigns as Mexico has now failed to qualify for quarter-finals for the fifth successive time. Mexico had an impressive run in the qualifying stage under Aguirre and were tipped to make it till quarter-finals but their defeat against Argentina cut-short the expectations of the fans and former players.
"It's very clear. The first person responsible is me. I believe that I have to resign my job," Aguirre told in a news conference.
Several reports suggests Aguirre was one of highest paid national team coaches with an annual salary of about $4 million. Only Italy coach Marcelo Lippi and England coach Fabio Capello are believed to earn more.
He had also coached the team in the 2002 World Cup, but was replaced after Mexico was eliminated by the United States.
He also made a statement that the new coach whosoever will replace him should be retained for the entire four-year World Cup cycle, which will end with the 2014 tournament in Brazil.