Al-Bashir is wanted by the International Criminal Court, where he faces charges of war crimes, crimes against humanity and genocide for his deadly campaign against insurgents in Darfur.
Meanwhile, Mr Zein Abedeen sought to reassure protesters who, while celebrating al-Bashir's removal, oppose the military's seizure of power in his wake.
The demonstrations against Bashir's 30-year rule first erupted in December, triggered by a tripling of the bread prices in one of the world's most impoverished countries.
The ousted president had come to power in a coup in 1989 and ruled Sudan for three decades with an iron fist.
"We, as a military council in our term, will not hand the president over overseas".
At the protest site itself thousands offered Friday prayers earlier in the day. The power of the people that brought down one of the longest serving despots in Africa is likely to send shivers throughout the rest of the Eastern and Horn of Africa region, whose people have been buckling under the misrule of strongmen, rigged elections and dwindling standard of living.
Zain al-Abideen said the military council would not interfere with a civilian government.
Thousands defied a warning from the military council to respect the night-time curfew imposed from 2000 GMT (10 pm) to 0200 GMT (4 am), to maintain their vigil outside army headquarters in Khartoum for a sixth straight night.
The Sudanese army claims it has no ambition to hold the reins of power, saying it responded to calls from the people against Bashir's rule.
Why now? Bashir was removed after months of anti-regime protests, with the military abandoning him and siding with those seeking his downfall.
Auto horns sounded on the streets as jubilant crowds streamed out of their homes to cheer the departure of Ibn Ouf, considered a regime insider and close aide of Bashir. Algeria's protest movement has been overwhelmingly peaceful and driven by young people frustrated with corruption and unemployment and who want a new generation of leaders to replace people like Bouteflika, ailing and hobbled since a 2013 stroke.
The Sudanese military has since taken over power and has announced that it will hand over back to a civilian leader after two years.
The Sudanese Professional Association (SPA), a leading group in the protest campaign that began late a year ago against Bashir, called ibn Auf and Mahi's resignation a "victory for the will of the masses", but it urged protesters to remain in the streets.
The force draws its origins from the Janjaweed militias that were implicated in the Darfur genocide.
Dozens of members of a paramilitary group stood at the sidelines, many atop pick-up vehicles loaded with machine-guns, as cheering crowds drove past, witnesses said.
In his televised announcement Thursday, Ibn Ouf said also that the military had suspended the constitution, dissolved the government, declared a three-month state of emergency and closed the country's borders and airspace.
"This is not a military coup but taking the side of the people".
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