Islamists have captured large swathes of the district in the southern province of Helmand which British and US forces struggled for years to defend.
Fleeing local residents reported bloody gunfights as the Taliban advanced on the district centre, highlighting a worsening security situation across Afghanistan a year after NATO formally ended its combat operations.
"We are air-dropping food supplies, military equipment and ammunition to support our forces in Sangin," defence ministry spokesman Mohammad Radmanesh told AFP.
A resident who fled Sangin said insurgents had publicly executed at least three security officials after storming government buildings.
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"The Taliban dragged two intelligence officials and a local police commander from their homes and shot them dead," Haji Abdul Qader said.
"Only the governor's compound and the police headquarters are under government control. The rest have been overrun by the Taliban."
Qader said he fled to the Helmand provincial capital Lashkar Gah after a mortar bomb landed on his house, wounding his infant son and daughter.
Taliban death squads were accused of summary executions, rape and plundering Kunduz as NATO-backed Afghan forces struggled for two weeks to evict them.
Highlighting the gravity of the situation in Sangin, long seen as a hornet's nest of insurgent activity, US and British forces were recently deployed in Helmand to advise and assist Afghan forces. This month marks a year since the NATO mission in Afghanistan transitioned into an Afghan-led operation, with allied nations helping to train local forces.