The army has been fighting the paramilitary Rapid Support Forces (RSF) for control of the capital and several major cities since April 15.
Attempts to mediate have proven fruitless as diplomats say both sides still believe they can win.
The fighting, during which neither side has gained a clear advantage, has inflicted high civilian casualties and displaced more than 4.5 million according to the United Nations.
In the video, which the army said was taken on Thursday in the Wadi Sayidna air force base in Omdurman - across the Nile from the capital - Burhan greeted cheering soldiers.
"The work you are doing should reassure people that the army has men and that Sudan is being protected by the army," he said in the video.
The video appearance comes after the RSF's multi-day attack on the Armoured Corps base in southern Khartoum, the army's only remaining stronghold in the capital outside its headquarters.
It was not clear how Burhan was able to leave Khartoum but analyst Cameron Hudson of the Center for Strategic and International Studies said it suggests the RSF may have taken a hit in costly battles, allowing the army greater movement.
The Wadi Sayidna base in Omdurman where Burhan appeared remains well protected despite RSF attacks.Â
Clashes between the forces continued in the city as the army works to cut off a key RSF supply route on the Shambat bridge to Bahri.
General Mohamed Hamdan Dagalo, head of the RSF, has often mocked Burhan for what he termed hiding in a bunker although Dagalo has only been seen in video once since the beginning of the war, speaking to soldiers outside a house in an unidentified location last month.
Mustafa Mohamed, an RSF advisor, accused Burhan of fleeing Khartoum in an interview with Sky News Arabia on Thursday, claiming the RSF had seized the Armoured Corps and would soon overrun army headquarters.
The army denied the RSF claims on the Armoured Corps in a statement earlier this week.
