Parwina Azimi, a women's rights activist in Sar-e-Pul, told AFP by phone that government officials and the remaining forces had retreated to a barracks about three kilometres (two miles) from the city.
"A plane came... but could not (land)," she said.
Kunduz, however, is the most significant Taliban gain since the insurgents launched an offensive in May as foreign forces began the final stages of their withdrawal.
It has been a perennial target for the Taliban, who briefly overran the city in 2015 and again in 2016 but never managed to hold it for long.
The ministry of defence said government forces were fighting to retake key installations.
"The commando forces have launched a clearing operation. Some areas, including the national radio and TV buildings, have been cleared of the terrorist Taliban," it said in a statement.
Kabul's inability to hold the north may prove crucial to the government's long-term survival.
Northern Afghanistan has long been considered an anti-Taliban stronghold that saw some of the stiffest resistance to militant rule in the 1990s.
The region continues to be home to several militias and is also a fertile recruiting ground for the country's armed forces.
US air strikes
On Friday the Taliban seized their first provincial capital, Zaranj in southwestern Nimroz on the border with Iran, and followed it up a day later by taking Sheberghan in northern Jawzjan province the following day.
Fighting was also reported on the outskirts of Herat in the west, and Lashkar Gah and Kandahar in the south.
The pace of Taliban advances has caught government forces flatfooted, but they had some respite late Saturday after US warplanes bombed Taliban positions in Sheberghan.
"US forces have conducted several air strikes in defence of our Afghan partners in recent days," Major Nicole Ferrara, a Central Command spokesperson, told AFP in Washington.
Sheberghan is the stronghold of notorious Afghan warlord Abdul Rashid Dostum, whose militiamen and government forces were reported to have retreated to the airport.
Dostum has overseen one of the largest militias in the north and garnered a fearsome reputation fighting the Taliban in the 1990s -- along with accusations his forces massacred thousands of insurgent prisoners of war.
Any retreat of his fighters would dent the government's recent hopes that militia groups could help bolster the country's overstretched military.
The government has said little about the fall of the provincial capitals, other than vowing they would be retaken.
That has been a familiar response to most Taliban gains of recent weeks, although government forces have largely failed to make good on promises to retake dozens of districts and border posts.
The withdrawal of foreign forces is due to be complete at the end of this month, ahead of the 20th anniversary of the 11 September attacks on the United States that sparked the invasion which toppled the Taliban.