The fighters said they had seized the arms cache from two military outposts in eastern Nuristan, abandoned by US forces last month.
Angela Eggman, a Nato spokeswoman, said it was not clear from the video where or when the weapons were obtained.
"Before departing the base, the units removed all sensitive items and accounted for them," she said.
But General Mohammad Qassim Jangulbagh, Nuristan's provincial police chief, disagreed, saying: "The Americans left ammunition at the base."
Farooq Khan, a spokesman for the Afghan National Police in Nuristan, concurred, saying US forces left arms and ammunition when they moved from the area, which he said was now in fighters' hands.
The Pentagon said the closing of the outposts in Nuristan was part of plans by General Stanley McChrystal, the US and Nato commander in Afghanistan, to shut down isolated units and focus on more heavily populated areas.
The developments come as Barack Obama, the US president, is due to meet military and national security advisers to discuss sending more troops to fight the Taliban in Afghanistan.
The White House has rejected a series of leaked reports saying Obama has already made up his mind to send thousands more troops to Afghanistan. It says no decision has been made.
Taliban fighters brandish what they say are US weapons seized from abandoned outposts
Meanwhile, Afghan police and Nato troops say they have seized a massive quantity of illegal fertiliser, enough to make hundreds of deadly roadside bombs, in the city of Kandahar.
A Nato spokesman said on Tuesday that raids at two sites in the southern city yielded more than 200 tonnes of ammonium nitrate - or about 10 lorry loads - and the arrest of 15 people.
Sunday's raids appeared to be one of the largest hauls of the war so far and Nato officials expressed hope that the seizure would hurt Taliban fighters, whose homemade bombs have become the biggest killer of foreign forces.
Acting on a tip, international forces and Afghan police discovered 1,000 45kg bags of ammonium nitrate fertiliser and 5,000 parts for roadside bombs in a warehouse.
An additional 4,000 45kg bags of fertiliser were found in a nearby compound soon after.
John Pike, director of the military think-tank Globalsecurity, said the seizure included enough fertiliser to make hundreds of roadside bombs.