Aguirre's Egypt was left stunned as the AFCON host lost 1-0 to South Africa in Cairo, the Pharaohs failing 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 after its group-stage exit at the 2018 FIFA 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 said after the disastrous defeat on Sunday (AEST).
"We are all upset about the early elimination and I bear all the consequences because I chose the players and the formation."