Gunman attacked the building after the blast, in the city of Mazar-e-Sharif, battling with German and Afghan forces.
The Taliban said it carried out the attack in retaliation for air strikes last week in Kunduz province.
Local police said at least one suspect had been arrested in connection with the bombing.
'Massive damage'
A Nato spokesman said there was "massive damage" to the consulate after the truck smashed into a perimeter wall.
