Afghan security officials said government forces had withdrawn in attempts to avoid civilian casualties and planned a counteroffensive seeking to regain Kunduz - a city that has already been the target of Taliban attacks twice this year.
"We are prepared and measures have been taken to recapture the city," Deputy Interior Minister Ayoub Salangi told reporters.
Both Afghan government leaders and the United States-led coalition view the defence of Kunduz as a key test of whether security forces can prevent the Taliban from expanding its reach in the country.
One Afghan official said Taliban fighters had control of all major government buildings in the city and security forces had retrenched to try to defend the airport of the provincial capital, about 240km north of Kabul.
"The Taliban have taken key government buildings such as the police and intelligence headquarters and burned down some of them," said Amruddin Wali, a member of the provincial council.
The assault appears to be the first time in the 14-year Taliban insurgency that large groups of fighters managed to penetrate deep into a major Afghan city with significant ground forces rather than carrying out isolated strikes and suicide bombings.
A statement from the Taliban said it would not seek retribution against local police or military officials.
The attack displayed the Taliban's battlefield power and co-ordination.
"This will have a lot of impact on morale on all sides," said Attiqullah Amarkhail, a retired Afghan general and military analyst. "Government forces may lose morale while opposition forces' morale will be boosted." But he noted Taliban gains did not necessarily foreshadow "the fall of the entire north or the fall of the Government".
Dominic Medley, spokesman for the United Nations Assistance Mission in Afghanistan, said all UN staff were evacuated from the area as security deteriorated.
Taliban fighters also overran the jail and freed hundreds of prisoners. Some inmates were seen walking down streets with their belongings.
In June, the Taliban briefly gained control of two of the city's six districts. Within days, however, Afghan security officials had regained control of the areas.