"Our forces completely destroyed the enemy's fighters and equipment, and seized large quantities of equipment and weapons," army spokesman Nabil Abdallah said in a statement broadcast on state television.
Abdallah vowed the army would "continue to progress on all fronts until victory is complete and every inch of our country is purged of the militia and its supporters".
On social media, soldiers shared videos appearing to be inside the presidential palace, exchanging congratulations. AFP could not immediately verify the footage.
Paramilitary forces occupied the palace in April 2023, when war broke out between the RSF and the army.
At the time, the RSF swiftly took over the streets of Khartoum, with the army-aligned government fleeing to Port Sudan on the Red Sea coast.
Central Khartoum, where the presidential palace stands alongside ministries and the capital's business district, has been the site of fierce battles in recent months, after army troops surged through the city.
Earlier this week, the army said its forces had merged from the north and south, hemming in the RSF.
In nearly two years, which has killed tens of thousands of people, uprooted over 12 million and created the world's largest hunger and displacement crises.
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