Despite Bashir's removal on Thursday, demonstrators had refused to disperse, camping out outside the army headquarters here, defying curfew declared by the military.
After Ibn Auf stepped down, the protesters celebrated his departure, with people chanting phrases like "it fell again".
The military council has also imposed a three-month state of emergency, with the constitution suspended.
Police said at least 16 people were killed by stray bullets at protests on Thursday and Friday.
Bashir's downfall followed months of unrest that began in December over rising prices.
The Sudan Professionals Association, which has been spearheading the protests, said Ibn Auf's decision to step down was a "victory" for demonstrators.
They were 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. The military council had said it would not extradite Bashir to face the ICC charges, which he denies.
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