Azam captained Pakistan in all three formats starting in October 2019 and led them to number one in the ICC One Day International (ODI) rankings for the first time in May.
Pakistan slid from the spot after finishing fourth in the Asia Cup and fifth in the World Cup.
"It's a difficult decision but I feel it is a right time for this call," Azam wrote on X, formerly known as Twitter.
"I will continue to represent Pakistan as a player in all three formats. I am here to support the new captain and the team with my experience and dedication."
His resignation comes two days after Pakistan bowling coach Morne Morkel stepped down.
The Pakistan Cricket Board (PCB) accepted his resignation. "After consultation with his family, Babar decided to step down and PCB stands behind his decision," they said in a statement.
Batsman Shan Masood has been announced by the PCB as the new Test captain while pacer Shaheen Afridi will be the T20 captain.
Azam is ranked by ICC as a top-five batsman in all three formats and is Pakistan's most successful captain in T20 Internationals with 42 wins.
