Aguirre's Egypt were left stunned on Saturday as the AFCON hosts lost 1-0 to South Africa in Cairo, where the Pharaohs failed to reach the quarter-finals for the first time since 2004.
Afterwards, Egyptian Football Association (EFA) president Hany Abou-Rida announced his resignation and confirmed the departure of the entire technical and administrative staff.
Despite his resignation, Abou-Rida – who also called for members of the board to step down – said he will continue in his position as head of the organising committee of the 2019 AFCON.
Egypt appointed Mexican boss Aguirre following their group-stage exit at the 2018 World Cup in Russia.
Former Japan and Mexico coach Aguirre signed a four-year deal to replace Hector Cuper last year.
"Of course I am responsible," the 60-year-old reporters post-match on Saturday.
"We are all upset about the early elimination and I bear all the consequences because I chose the players and the formation."