The decision comes despite Hassan guiding the Pharaohs to the round of 16 at this year's World Cup, where they recorded their first-ever victory in the tournament.
The 60-year-old had earned a contract extension following Egypt's impressive campaign, which included a victory over New Zealand before they eliminated Australia on penalties in the round of 32.
However, their World Cup journey ended in the last 16 after surrendering a lead to suffer a 3-2 defeat against Argentina.
In a statement released on Saturday, the EFA confirmed that a review of recent events had led to the decision to make changes to the technical team.
"The Egyptian FA thanked Hossam Hassan for his efforts during his time in charge and wished him success in the next stage of his career. The board also appointed Ali Maher as head coach of the national team for the coming period," the statement read.
Tensions had also emerged following Egypt's 0-0 home draw against Angola in a September Africa Cup of Nations qualifier, particularly after Hassan substituted captain Mohamed Salah in the 60th minute.
Salah subsequently left the national team camp, with the EFA explaining that the Trabzonspor forward avoided playing on artificial pitches and would miss the fixture against South Sudan.
EFA president Hany Abo Rida later travelled to Turkey to hold discussions with the Egyptian captain.
The 52-year-old Maher now takes over following his spell in charge of Ceramica Cleopatra.
Hassan departs after 38 matches, recording 21 victories, 10 draws and seven defeats. He had also received a three-match FIFA ban following his criticism of the referee after the Argentina defeat.
Egypt are scheduled to face Malawi twice this month in their upcoming AFCON qualifiers.