A soldier from the US-led coalition was also wounded in the operation on Thursday in Uruzgan province, the US military said.
Coalition and Afghan troops came under attack from up to 20 rebels "firing small arms, heavy machine guns, mortar rounds and rocket-propelled grenades," the US military said in a statement on Friday.
"Coalition and US close air support and US attack helicopters arrived at the scene, blasting enemy positions killing 10 enemy combatants," it said.
Uruzgan governor Jan Mohammed Khan said the operation was launched after a tip-off that the man considered a military chief of the Taliban, Dadullah, was hiding out in the province's Charchino district.
"We had reports that Dadullah was hiding in the area and we launched an operation but we faced Taliban resistance and the fighting broke out," he said.
"Coalition and US close air support and US attack helicopters arrived at the scene, blasting enemy positions killing 10 enemy combatants"
US military statement
A purported spokesman for the Taliban said six of the group's fighters were killed. Abdul Latif Hakimi also claimed that eight US and 10 Afghan soldiers were killed but the US and Afghan armies said only one Afghan had died.
They would not confirm the nationality of the coalition soldier wounded in the clash.
Australia's defence department said earlier on Friday that an Australian special forces soldier was wounded in an operation in Afghanistan in which an Afghan soldier was killed.
In a statement released in Australia, the department did not say when or where the clash took place but said the soldier was already back on duty.
This year has seen the worst rebel-related violence in Afghanistan since the Taliban were toppled. More than 1000 people have been killed, including 50 US troops, the most since the coalition arrived in 2001.
Meanwhile, a band of seven wedding singers was shot and killed on their way home from a late night performance in northern Afghanistan, police said on Friday.
The men were ambushed as they were returning to Mazar-i-Sharif, the capital city of northern Balkh province, after a show in Chimtal district about 50km away, police officer Mohammed Azam said.
"Unknown armed men ambushed their vehicle and opened fired at them and killed them. It doesn't seem that the attackers were thieves because they did not steal anything, even the car," Azam said.
"They might be personal enemies of the seven or some extremist group, but it is premature to tell," he said.
Sparsely populated Balkh is usually free of the Taliban-linked violence that plagues southern Afghanistan but is home to various rival warlords.