There was no immediate claim of responsibility for the attack.
Khost, which is on the border with Pakistan, has been reeling from a series of attacks claimed by Taliban fighters.
The Taliban, whose members were in government from 1996 until the 2001 US-led invasion, are leading an armed campaign supported by fighters across the border in Pakistan.
The attack in Khost coincided with fighting in which 30 suspected Taliban-linked fighters were killed in a police operation in southern Afghanistan.
Three others died in an explosion during a bomb-training session, the authorities said on Tuesday.
The interior ministry said in a statement the operation in the south was launched on Monday in Uruzgan province and continued on Tuesday in neighbouring Helmand, a stronghold of Taliban activity.
"During this operation 30 enemies of peace and stability were killed and 17 were injured," the statement said.
It said Mullah Mohammad Yaqoub, a "famous Taliban commander", was among the dead.
Weapons were also seized during the fighting, one of the biggest raids against the fighters in recent weeks, the ministry said.
Mohammand Gulab, Uruzgan's deputy police chief, said that 150 Afghan police took part in the operation and were assisted by international troops. Four policemen were wounded in the fighting, he said.
The US, Britain and Nato have more than 70,000 troops in Afghanistan fighting the Taliban.