Although the country remains in a deep economic crisis, there had been more international support – and a military takeover will put that hugely at risk.
One demonstrator, Haitham Mohamed, told Agence France-Presse: “We are ready to give our lives for the democratic transition in Sudan.”
Prime Minister Abdallah Hamdok is among those reported to have been put under house arrest, along with members of his cabinet and other civilian leaders.
A statement from the information ministry on Facebook said those arrested were being held in “an unidentified location”.
It also said Mr Hamdok was being pressed to support a coup but was refusing to do so and he urged people to continue with peaceful protests to “defend the revolution”.
Gen Burhan has been heading the Sovereign Council, part of a power-sharing arrangement between military and civilian leaders.
In a televised address, he said infighting between politicians, ambition and incitement to violence had forced him to act to protect the safety of the nation and to “rectify the revolution’s course”.
He said Sudan was still committed to “international accords” and the transition to civilian rule, with elections planned for July 2023.
But thousands of protesters have already taken to the streets in Khartoum to denounce a coup. Some chanted “no to military rule”.
Demonstrator Sawsan Bashir told AFP: “We will not leave the streets until the civilian government is back and the transition is back.”
Witnesses say the internet is down and that army and paramilitary troops have been deployed across the city. Khartoum airport is closed, and international flights are suspended.