The bomber died in the attack, which also left two civilians injured, at the entrance to Bagram Airfield, Afghan Interior Ministry spokesman Sediq Seddiqi said.
A spokesman for the International Security Assistance Force identified the two injured people as members of the NATO-led ISAF.
CNN quoted a Taliban spokesman as saying 12 U.S. troops and nine Afghans have died in the latest chapter of its "revenge campaign."
A U.S. investigation determined the Koran burnings last month were inadvertent and the result of the books having been defaced by prisoners attempting to get messages to each other.
U.S. President Barack Obama apologized and promised disciplinary action against those responsible.
At least 30 Afghans have been killed in protests over the Koran burning, and officials have speculated some of the six killings of U.S. military personnel since then were motivated by outrage over the incident.
Kate Middleton recycles dress at movie premiere
Gal Gadot cast as Wonder Woman for 'Batman vs. Superman'