“Nine insurgents have been killed, one other is injured and another one is still holding up against our forces,” the ministry said.
Taliban fighters stormed a perimeter section of the heavily fortified site which contains both a civilian airport and the joint Afghan-NATO military base.
Samim Khpalwak, a spokesman for the Kandahar provincial governor, said Taliban gunmen were targeting residential blocks housing government employees and military.
“Several insurgents managed to breach the first gate of the complex,” he said.
“They have taken up position in a school inside the complex.”
The provincial governor’s office had previously put the death toll at nine.
The government claimed that an unknown number of assailants had been killed but residents were still reporting a volley of gunshots and explosions.
“The fighting started around 6:00 pm last night and intensified over the night,” said 30-year-old university student Izatullah, who lives inside the complex.
“Soldiers were calling on Taliban attackers to let women and children go but attackers declined. We could hear children screaming during the fighting.”
Airline passengers were also trapped inside the civilian terminal, far from the fighting in the sprawling complex, when their commercial flight to India was suspended, Kandahar airport director Ahmadullah Faizi said.
A statement from the Taliban said suicide attackers armed with both light and heavy weapons had entered the base in Kandahar and had attacked international forces and their Afghan allies.
It said 150 soldiers had been killed, but there was no official confirmation and the insurgents often made exaggerated casualty claims.
The attack came ahead of Afghan president Ashraf Ghani’s visit to Pakistan for a regional conference on peace talks with the Taliban.
The Taliban group has been largely absent from cities since being driven from power by the US and its allies, but has maintained an often brutal rule over swathes of the countryside