It marked the fifth straight time that Mexico has been eliminated in the second round of the World Cup.
Aguirre said on Wednesday that he was "the person responsible" for the loss.
Mexico has only twice reached the quarterfinals of the World Cup -- in 1970 and 1986, when it hosted the event.
Aguirre made the announcement at a packed news conference in Mexico City. The Mexico-born Aguirre was appointed 15 months ago for a second stint as Mexico coach. He also coached the team in the 2002 World Cup, but was replaced after Mexico was eliminated by the United States.