The militants began their assault on the Qala-e-Zal district in Kunduz province on Saturday and captured most of it including the district centre.
The attack triggered intense fighting with government forces that sent civilians fleeing toward the provincial capital of Kunduz.
The Norwegian Refugee Council, which has offices in Kunduz, said in a statement that some families were forced to sleep in the open while others moved into the homes of relatives.
"Taliban fighters were shooting from one side of our house and the Afghan National Security Forces from the other side. A rocket landed in my garden. My wife told me that if I didn't flee, she would leave the house alone with our two daughters," said another resident, 29-year-old Rahman Gerdi.
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Mahfuzullah Akbari, a government spokesman, said Afghan forces retreated from the district to avoid civilian casualties while a Taliban spokesman said in a statement they had forced them out.
On April 28 the Taliban launched their "spring offensive", heralding a surge in fighting as the US tries to craft a new Afghan strategy and NATO considers sending more troops to break the stalemate against the resurgent militants.
The annual spring offensive normally marks the start of the fighting season, though this winter the Taliban continued to battle government forces. An attack on a military base in the neighbouring city of Mazar-i-Sharif left at least 135 soldiers dead earlier last month.
The militants also claimed to have captured Sangin district in the volatile southern province of Helmand in March, underscoring their growing strength.