The assailant had a second grenade and two landmines hidden on his body when he approached Camp Salerno, an American base in eastern Khost province, asking for medical help on Friday, the US military said in a statement on Sunday.
He was arrested after pulling the pin on a grenade in his hand, but it failed to detonate.
The last such assault was in June, when a bomber killed 20 people at the funeral of a pro-government cleric.
The fighting at the Taliban camp occurred in Zabul province's Shahr-e-Safa district on Saturday and was between rebels and American and Afghan forces, said the district's chief, Ghulam Rasol.
He said that besides the eight insurgents killed, an Afghan soldier was also wounded.
However, US military spokeswoman Lieutenant Cindy Moore said the only report she had of fighting in the area on Saturday was Afghan police killing a fighter after coming under attack. She said US and Afghan forces pursued the insurgents, who fled after ambushing the police, but failed to catch them.
In other fighting, Afghan and US forces battled suspected Taliban rebels in mountains in Uruzgan province, killing an insurgent and arresting seven others, including a regional commander, said provincial Governor Jan Mohammed Khan.
In another development, US forces and police destroyed three bombs found separately in eastern and southern Afghanistan on Thursday and Friday, another military statement said.