The clash happened in the southeastern province of Paktika on Tuesday after Afghan and US troops encountered the rebels during a patrol, a statement issued late on Wednesday said.
"The enemy fled shortly afterwards toward a nearby cave complex," it said.
"Afghan and US forces pursued the enemy combatants toward the cave complex killing one," it said, adding that several hours later the same forces came into contact with additional fighters and killed five more.
There was no immediate comment from the Taliban on the incidents, but spokesman Abdul Latif Hakimi said the fighters had killed two Afghans in the central province of Uruzgan for "spying" for the Americans.
Meanwhile residents of a remote village in southern Afghanistan said on Thursday that US warplanes bombed houses during its operations this week, killing several civilians and wounding others - including a six-week-old baby.
Zabul province Governor Ali Khail said US-led forces had "made a mistake" during recent operations in the province, but gave no details. "There were some casualties among villagers, but I don't know the exact number of casualties," he said.
The US military, however, denied any civilians had been in the Day Chopan district where the fighting had taken place on Monday.
The military said earlier that 18 suspected rebels and one US service member were killed in the fighting - one of five Americans to die in action in Afghanistan in the past week.
"The children were crying and they were very afraid. These planes killed my relatives. We are poor and innocent people. Why are they killing us?"
Resident of the bombed village
Two villagers spoke to The Associated Press at a hospital in the provincial capital, Qalat, a few hours drive from their home village of Rauf, which they said had been pounded by American forces on Monday night and early Tuesday.
"The children were crying and they were very afraid," said a weeping Sadia Bibi, 50. "These planes killed my relatives. We are poor and innocent people. Why are they killing us?"
Bibi's 20-year-old daughter Najiba Bibi and six-week-old grandson were being treated at the hospital for injuries to their hands and legs, which she said had been struck by pieces of brick during the bombing. Both the woman and boy were bandaged.
Bibi said her 55-year-old brother, Abdul Shakor, and his wife were killed along with a 16-year-old boy named Matiullah.
A relative who brought the injured to the hospital, Abdul Halim, 35, said his neighbour's house had been bombed, killing a man who lived there.
Also, one woman from the village died at a hospital in neighbouring Kandahar province after arriving there with two other injured women on Wednesday, a doctor at the hospital, Mohammed Hashil, said.
US military spokeswoman, Lieutenant Cindy Moore, however, said intelligence indicated there had been no civilian casualties.
"My understanding is that our intelligence shows no civilians in this area. It was a remote area. The targets were in an open area. We were firing back ... this is possibly propaganda press. We don't have any assessment of any civilians in this area," she told The AP.
Afghan elections are planned
for 18 September
The US military reported earlier that the fighting in Day Chopan was sparked when a US and Afghan patrol came under fire from rebels using small-arms and rocket-propelled grenades, and that coalition aircraft had joined the battle.
Afghan officials and human rights groups have complained repeatedly about civilian casualties in US-led military operations, saying heavy-handed tactics could raise sympathy for fighters who have maintained their resistance to Karazai's government since the US invasion in 2001.
American commanders insist they modify their operations to try to avoid hurting civilians and accuse rebels of using civilians for protection.
Sporadic attacks across the country have deepened concerns over security ahead of key legislative elections set for 18 September.