UN Secretary-General Antonio Guterres appealed for "calm and utmost restraint by all" and urged a transition that would meet the "democratic aspirations" of the people.
But the protesters' jubilation quickly turnedto anger when Ibn Auf, who Bashir appointed first vice president in February, also said that the military council would run the country for a two-year transitional period.
Pressure increased this week, as protesters held a sit-in outside army headquarters in Khartoum.
As a united force, Sudanese citizens from diverse backgrounds joined the protest - dubbed the Sudanese Revolution - which started on December 19, 2018, demanding the resignation of the president as a result of the increasing costs of living in the country beleaguered with a cash crisis for a better part of the year. The military has also reportedly taken over state media and curtailed operations at the Khartoum airport.
The SPA said the military had announced a "coup" that would merely reproduce the same "faces and institutions that our great people revolted against".
"[There are] no services, no life, and the regime would repeat lies and false promises and implement their heavy-handed security solution to the people", Mr Auf said in his address.
Al-Bashir governed with an iron fist means while also appearing to offer concessions - a tactic created to divide his opponents.
"We finally win this battle, we struggled a lot and we suffered a lot but everything (is) supposed to have an end", 45-year-old tea seller Fathia Imam told Al Jazeera at the square of the sit-in.
But the festive mood later soured.
Earlier Thursday, the Sudanese army announced the "removal" of al-Bashir and the imposition of a two-year "transitional phase".
"We are not leaving, we are not leaving". They danced and shouted to celebrate.
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Formerly an army officer, he seized power in a military coup in 1989.
United Nations chief Antonio Guterres called for a transition that would meet the "democratic aspirations" of the Sudanese people and appealed for "calm and utmost restraint by all", his spokesman said.
In recent days, soldiers protected demonstrators from other security services that were attempting to disperse them.
The military council also said it was declaring a ceasefire across the country, including in war-torn Darfur.
Over his three decades in control, he was forced to allow the secession of South Sudan after years of war, a huge blow to the north's economy.
South Sudanese rebel leader Riek Machar said in Rome he hoped the situation would be handled "peacefully so that the country can be stable".
The State Department said the airport in Khartoum will be closing and that the the consular section at the US embassy in Khartoum was also closed Thursday.
Names of Sudan's possible successors that have been circulating include the Defence Minister, an ex-military intelligence chief, also an Islamist, and former army chief of staff Emad al-Din Adawi.
But in the eastern cities of Kasala and Port Sudan, the releases failed to materialise, prompting protesters to storm NISS buildings, according to witnesses.
The latest crisis has escalated since the weekend when thousands of demonstrators began camping out outside the defence ministry compound, where Mr Bashir's residence is located.
Government officials said 38 people had died since December but Human Rights Watch said the number was higher.
When he was indicted by the International Criminal Court in 2009 on charges of committing crimes in Darfur, he responded by expelling a dozen aid groups working in the war-plagued region, where up to 300,000 people were killed and 2.7 million driven from their homes by militias he backed.