The dead are the latest victims of a stubborn insurgency that undermines the country's fragile democracy.
The first bomb exploded shortly after midnight as a convoy of troops was returning from an operation in Kunar province.
It killed two soldiers and wounded five, said Mohammed Hasan, the regional deputy police chief.
Hours later, as other troops were rushing to the site of the first blast, a bomb hit their vehicles and six soldiers were killed and one was wounded, he said.
"The situation is very bad. We've come under attack throughout the night," Hasan said.
The dirt road where the attack occurred is often travelled by Afghan and US forces and is a frequent site of bombings and ambushes.
US troops often use bomb-clearing machines to make it safe before sending convoys down it.
Hasan said his forces lack the equipment.
Kunar is a province of steep mountains, winding valleys and numerous caves - terrain favoured by insurgents.
It adjoins the border with Pakistan and is suspected to be a hideout for leaders of al-Qaida and other groups.
The deadliest attack on US forces in the past year occurred there, when suspected Taliban fighters last June killed three American commandos on the ground and then shot down a special forces helicopter, killing 16 troops.