The speech came after the resignation of feared security chief Gen Salah Gosh hours after the coup leader himself, Defence Minister Awad Ibn Auf, stepped aside.
Ibn Ouf said: "I, the head of the military council, announce I am giving up the post", he said, adding he took the decision to preserve unity of the armed forces.
After denying charges of a military coup, Abdin said the transitional military council, due to stay in power for two years, would "open dialogue with political parties".
Burhan, who previously served as the general inspector of the armed forces, had met with the protesters in the streets after al-Bashir's ouster and listened to their demands.
"He's never been in the limelight like Ibn Ouf or General Kamal Abdelmarouf", the officer said, referring to the army's former chief of staff.
Sudan's army has ousted veteran president Omar al-Bashir, but protestors against his iron-fisted rule denounced a "coup" and thousands rallied outside army headquarters as a night-time curfew kicked in.
The umbrella group organising the protests called on newly sworn-in ruler Gen Burhan to "transfer the powers of the military council to a transitional civilian government".
They are demanding a transition to civilian rule before they return home.
The Trump administration has said it will withdraw or deny visas of any ICC judges or prosecutors if the court investigated possible war crimes against U.S. forces in Afghanistan.
"The stepping down of General Awad Ibn Ouf is a victory of the people's will", the Sudanese Professionals Association said. He has close ties to senior Gulf military officials as he was responsible for coordinating Sudan's military involvement in the war.
Amazon employees paid to listen to private Alexa chats
Google does the same for its personal assistant but distorted the audio before it is listened to by an employee. They were apparently told that they couldn't do anything about it, because it's not Amazon's job to interfere.
The military leaders who carried out a coup in Sudan on Thursday have sought to reassure people that their only concern is public order.
"The new government authorities of Sudan must surrender al-Bashir, as well as Ahmad Muhammad Harun, Ali Muhammad Ali Abd-Al-Rahman (Ali Kushayb"), Abdel Raheem Muhammad Hussein, and Abdallah Banda Abakaer Nourain to the ICC in order to provide justice to the victims of the worldwide crimes and the other perpetrators of the human right violations that have been committed in the country.
Sara Abdelgalil, a spokeswoman for the Sudanese Professionals Association, said the global community should now focus on preventing a massacre in Sudan.
Early on Thursday, military vehicles entered the large compound in Khartoum housing the defence ministry, the army headquarters and Mr Bashir's personal residence.
Al-Bashir's removal resembled similar military interventions against longtime rulers in other countries in the region during the so-called Arab Spring of 2011.
"We as a military council, we will not deliver the President overseas during our period" in office, Mr Abdin said when asked about the ICC arrest warrant.
Protest organizers have vowed not to end their street action until a civilian transitional council is formed, saying rule by military commanders who for years were al-Bashir loyalists is just an extension of his regime.
Zain al-Abideen said the military council would not interfere with a civilian government.
"Moreover, the suspension of the constitution could be lifted at any point and the transitional period could be shortened depending on developments on the ground and agreements reached between stakeholders", the Sudanese envoy said.