The fighting late on Monday in Uruzgan province's Dihrawud district occurred during an offensive against an insurgent camp, which had been used as a base for attacks in neighbouring areas, provincial governor Jan Mohammed Khan said on Tuesday.
About 25 suspected insurgents were captured, he said, adding that Afghan forces were still finding bodies of fighters on Tuesday.
US soldier killed
A US military statement on Monday said heavy fighting in the area had killed one American service member, an Afghan soldier and 11 insurgents.
Three US soliders and one Afghan soldier were also wounded, it said.
The statement said American fighter jets and helicopters responded to a rebel attack on a patrol that sparked the gun battle.
In fighting elsewhere, police arrested 10 suspected Taliban suspects after clashes in the southeastern Zabul province, said Ali Khail, a spokesman for the provincial governor.
In neighbouring Kandahar province, Taliban fighters attacked an Afghan patrol on Monday night, triggering a gun battle that left an Afghan soldier dead and a police officer badly wounded, said deputy district chief Haji Lala Khan.
In eastern Paktika province on Tuesday, a candidate in the upcoming legislative elections was killed when a roadside bomb blew up next to his vehicle as he was taking his mother to hospital, local police chief Malik Khan said.
The latest fighting comes three days after the US military operational commander in Afghanistan, Major-General Jason Kamiya, said that fighting in recent months had devastated the ranks of the Taliban.
More than 800 people have been killed in a major upsurge in violence since March, and US and Afghan officials have warned that it may threaten the parliamentary elections scheduled for 18 September.
Meanwhile, more than 1000 protesters on Tuesday chanted "Death to America" as they banged on the gates of the main US base in Afghanistan to demand the release of three local villagers detained in a raid.
The three men were arrested on Monday night in a raid on Dheymullah village, about 3km from Bagram base, said local government chief Kaber Ahmad.
"Those three were wanted by the Americans, but no one knows why," he said.
A US military spokeswoman in Kabul said she had no details about the demonstration.