The fighting on Saturday, which also wounded a US service member and four Afghan soldiers, was the deadliest in recent weeks and comes a month after landmark legislative elections that many people had hoped would sideline the fighters.
Violence also broke out on Saturday in northern Afghanistan, which has been spared much of the bloodshed suffered in other areas, when armed men fired at a patrol of British peacekeepers in Mazar-e-Sharif city and wounded four of them, said Sheir Jan Durani, a police spokesman.
A spokesman for a Nato-led peacekeeping force, Captain Michele Cortese, confirmed that four British members of the force had been wounded. Security forces cordoned off the area and arrested four suspects.
Police and peacekeepers in Kabul discovered a large weapons cache in an old building, including rocket-propelled grenades, anti-tank missiles, bombs and ammunition, according to a statement by the Nato force.
In the latest fighting, a US paratrooper was killed on Saturday after his patrol came under fire in Khost province, near the eastern border with Pakistan, a US military statement said.
American forces responded with small-arms fire, artillery and air attacks, prompting the militants to flee. An assessment of the battle was ongoing and it was not immediately clear whether any of the assailants was killed, it said.
The death brought to 203 the number of US soldiers killed in and around Afghanistan since the Taliban was ousted in 2001.
The deadliest fighting, however, was in southern Uruzgan province on Thursday, a separate US military statement said. It started after a joint US-Afghan patrol was attacked by militants firing assault rifles, mortars and rocket-propelled grenades.
"A total of 13 enemy fighters were killed in the three engagements"
A US military statement
A US service member and an Afghan soldier were wounded in this assault.
Shortly afterwards, militants launched a second attack a few kilometres from the first, killing an Afghan soldier and wounding three others. The wounded were evacuated to a hospital at a nearby base. There were no US or Afghan casualties in the third battle.
"A total of 13 enemy fighters were killed in the three engagements," the statement said. "Coalition aircraft and attack helicopters provided close air support for the operations."
In eastern Paktika on Friday, American troops attacked a group of militants as they laid a roadside bomb, capturing two and killing one as he tried to flee, a third US statement said. A fourth rebel managed to escape.
Two mirrors used for signalling to other fighters and blasting caps were found on the slain man.
Elsewhere, suspected Taliban rebels fired at a vehicle late on Friday in southern Helmand province and killed two brothers and a son of one of the men, said Ghulam Muhiddin, a local government leader.
Taliban rebels are said to have
stepped up attacks this year
It was not clear what motivated the attack. One of the men was a candidate in legislative elections last month but had lost, Muhiddin said.
Fighters also attacked Muslim worshippers, dragging two men from their prayers in a mosque in eastern Paktia province before killing both outside, and shooting dead a tribal elder as he prayed in neighbouring Khost province, officials said.
Taliban-led rebels have stepped up attacks this year, leaving almost 1500 dead.
The violence is the deadliest since US-led forces ousted the Taliban government from power in late 2001 and has raised fears for the country's fragile democracy.