Police and US military on Friday said the US soldier was killed and nine others wounded by a mine near Ghazni.
In Khost, at least two Afghans were killed and six wounded in an explosion at a government military post near the former Taliban stronghold in southeastern Afghanistan.
The blast came hours after more than 20 rockets landed near Khost's airport, where US-led forces are based.
It was the biggest single rocket attack since the fall of the Taliban in late 2001, but US military spokesman Lieutenant-Colonel Bryan Hilferty reported no casualties.
"There were no coalition casualties," he said in a statement.
US-led forces returned fire with artillery and precision-guided bombs but "at this time there is no indication of anti-coalition militia casualties," Hilferty said.
The attacks in Khost are the latest in a wave of violent incidents which have killed more than 550 people in the last six months predominantly in southern and eastern Afghanistan.
Khost's deputy intelligence chief was shot dead only two days ago by a suspected Taliban fighter, who then blew himself up to avoid arrest.