Ramazan Bashardost, a former planning minister and one of 41 candidates in the presidential race, said later that he was not the target of the attack.
A Taliban spokesman confirmed to Al Jazeera that Bashardost had not been expressly targeted.
An Afghan official said the main targets appeared to be the police headquarters, attorney general's office, courts and a local bank branch.
The attack on Khost comes days after a similarly brazen assault by Taliban fighters and suicide bombers, some dressed as women, on the eastern city of Gardez in which nine people were killed.
Afghanistan has scheduled elections for August 20, but with growing violence across the large parts of the country and Afghan and international forces seemingly unable to ensure security, there are fears that few voters will turn out.