Defence Minister Awad Ibn Auf announced his decision on state TV. He named as his successor Lt Gen Abdel Fattah Abdelrahman Burhan.
It comes after protesters refused to leave the streets, saying the coup leaders were too close to Bashir.
The army has said it will stay in power for two years, followed by elections.
Bashir’s downfall followed months of unrest that began in December over rising prices. At least 38 people have died in the protests.
Ibn Auf was head of military intelligence during the Darfur conflict in the 2000s. The US imposed sanctions on him in 2007.
Protesters in Khartoum celebrated his departure.
The Sudan Professionals Association, which has been spearheading the protests, said Ibn Auf’s decision to step down was a “victory” for demonstrators.
They are demanding a transition to civilian rule before they return home.
Bashir has been indicted by the International Criminal Court (ICC) on charges of war crimes and crimes against humanity over the Darfur conflict.