Hopes were high for Morocco in the Ivory Coast after Regragui had led them to the semi-finals of the 2022 World Cup in Qatar, a first for any African nation.
After finishing top of their group with two wins and a draw, however, the Atlas Lions lost 2-0 to South Africa in the round of 16, which "caused disappointment", the federation said in the statement.
"I failed. It's my failure, not the players," said Regragui who had previously said he would step down if Moroccan did not at least reach the quarter-finals.
But the head coach will remain in place with the Moroccan federation saying it wanted to "mobilise all means to ensure full success for the national team" ahead of the 2025 AFCON and 2030 World Cup, which are both set to take place in the North African country.
Regragui's AFCON was also marred when he was suspended by the Confederation of African Football (CAF) after an altercation with Congolese defender Chancel Mbemba at the end of a group match - a decision that was later overruled after an appeal from the Moroccan federation.
