The Taliban claim, for which there was no confirmation from the Afghan government, comes after reports that Aghan President Ashraf Ghani fled to Tashkent, Uzbekistan.
The Taliban has taken control of Afghanistan's presidential palace, two senior Taliban commanders present in Kabul told Reuters, after President Ashraf Ghani left the country.
A Taliban commander said the group will soon declare the Islamic Emirate of Afghanistan from the presidential palace in Kabul.
Images broadcast on television showed dozens of Taliban fighters inside the presidential palace, declaring victory over the Afghan government.
"Our country has been liberated and the mujahideen are victorious in Afghanistan," one militant told news channel Al Jazeera from the palace.
The fighters showed reporters their weapons in a tour of the building, seized after President Ashraf Ghani fled the country.
Real test begins now
One of the Taliban's most senior officials declared on Sunday that the movement's swift victory over the Afghan government was an unrivalled feat but that the real test of governing effectively would begin now that it had won power.
In a brief video statement, Baradar, the head of the Taliban's political bureau, said the victory, which saw all of the country's major cities fall in a week, was unexpectedly swift and had no match in the world.
However, he said the real test would begin now with meeting the expectations of the people and serving them by resolving their problems.
READ MORE: Afghan President Ashraf Ghani flees country
US embassy suspends operations in Kabul
Meanwhile, the US Embassy in Kabul has suspended all operations and told Americans to shelter in place, saying it has received reports of gunfire at the international airport.
The US is racing to airlift diplomats and citizens out of Afghanistan after the Taliban overran most of the country and entered the capital early Sunday.
Earlier on Sunday, the insurgents captured the eastern city of Jalalabad without a fight, giving them control of one of the main highways into landlocked Afghanistan.
They also took over the nearby Torkham border post with Pakistan, leaving Kabul airport the only way out of Afghanistan still in government hands.
President Joe Biden on Saturday authorised the deployment of 5,000 US troops to help evacuate citizens and ensure an "orderly and safe" drawdown of military personnel.
Biden said his administration had told Taliban officials in talks in Qatar that any action that put US personnel at risk "will be met with a swift and strong US military response."
He has faced rising domestic criticism after sticking to a plan, initiated by his Republican predecessor Donald Trump, to end the US military mission in Afghanistan by August 31.
"An endless American presence in the middle of another country's civil conflict was not acceptable to me," Biden said on Saturday.