"Kunduz city has collapsed into the hands of the Taliban," Interior Ministry spokesman Sediq Sediqqi told the Associated Press.
The fall of Kunduz marks a major setback for government forces, who have struggled to combat the Taliban since the US and NATO shifted to a supporting role at the end of last year.
Military reinforcements have been sent to Kunduz, where government forces managed to fend off a major Taliban assault earlier this year. "We are trying our best to clear the city as soon as possible," Sediqqi said.
Residents said the militants reached the main square 12 hours after launching their attack. They said photographs of President Ashraf Ghani and other leaders were torn down and the white flag of the Taliban was raised. They said residents were streaming to the airport in an effort to flee.
Also Read
The deputy spokesman for President Ashraf Ghani had earlier described the situation in Kunduz as "fluid." Zafar Hashemi said the president was "in constant contact with the security and defense leadership to provide them with guidance."
The medical charity Doctors Without Borders said it had treated more than 100 wounded people in Kunduz since the assault began, including 36 who were "in critical condition, with severe abdominal and head injuries."
The Taliban used social media to claim the "conquest" of Kunduz and reassure residents that the jobs of teachers, doctors and other civil employees, and their personal property, were safe. The Taliban have a history of brutality, and are known to ban women's education as well as music, movies and other trappings of modern life in areas they control.