Fighting in the Sangin district of Helmand was continuing as army and police arrived to help security forces who had been pinned down for days, Masoom Stanekzai said.
Afghanistan's embattled security forces needed international military help, especially air support, which would help reduce casualties, Stanekzai told reporters.
Sangin is an important poppy-growing district in Helmand, which borders Pakistan and sits on transport routes for drugs, arms and other lucrative contraband.
"The Helmand battle is not easy because the province has a long border, is a core of opium production, and our enemies are well-equipped and deeply involved in the smuggling of drugs," he said. "These factors complicate the battle for Sangin."
Taliban spokesman for southern Afghanistan, Qari Yousaf Ahmadi, tweeted that "Sangin district has completely collapsed to the Taliban" by mid-afternoon Wednesday with the capture of soldiers and ammunition. The insurgent group is prone to exaggerating its battlefield successes, and Afghan officials denied Sangin had fallen.
However, Helmand's deputy governor Mohammad Jan Rasulyar said all lines of communication with Sangin had been cut and there was no immediate information available on the situation there, AP reported.
Britain has sent a small contingent of soldiers to Helmand as advisers under the NATO mandate. The return of British troops is poignant, as they suffered more than 100 of their 456 fatalities during Britain's 13-year Afghan combat mission in Sangin.
Afghanistan's security forces are taking on the Taliban alone, following the end of the international combat mission last year. The US and NATO have around 13,000 troops in the country, most of them operating under a training mandate.
Districts across Helmand have been threatened by the Taliban in recent months. The fight for Sangin has been particularly ferocious, with officials saying that only the army base was still in government hands until Tuesday.
Supply lines were cut, preventing ammunition and food from reaching government forces, and roads around the district center mined, officials have said.
Stanekzai said that with the arrival of new troops to the area, the battle would be reinvigorated and "this should help cut the number of casualties, and provide much-needed logistical support."
The Taliban have fought hard across the country this year, stretching Afghan forces as they have taken hold of some districts, if only for a few hours or, in the case of the northern city of Kunduz, three days that sent waves of concern through Afghans who had believed the insurgency was not strong enough to take major urban centers.