Kandahar's education chief and the province's deputy head of the health department were among the dead, Ahmad Wali Karzai, the council head and a brother of Hamid Karzai, the Afghan president, told reporters in Kandahar.
Sixteen people, including members of the council, were wounded, officials added.
Earlier, Zemeri Bashary, an Afghan interior ministry spokesman, said five civilians and a police officer had been killed.
He said three attackers had also been killed in the assault, which began with a car bomb explosion.
The attack bore similarities with a February assault in Kabul where fighters killed 20 people in simultaneous raids on government buildings.
Sayad Ahmad, a council official, said the attackers wore suicide vests.
"Three men with military-style uniforms started opening fire on our security guards, before, one by one, they detonated themselves," Ahmad said.
Haji Agha Lalai, a council member, told the Associated Press news agency he was inside the building when he heard the first explosion.
A surge in violence in Afghanistan is expected this year as 21,000 new troops deployed by Barack Obama, the US president, arrive.
Taliban-linked fighters have carried out a string of deadly attacks in recent months and now control large parts of the countryside where Afghan and international forces do not have enough manpower to maintain a permanent presence.
More than 70,000 US, British and Nato troops are in Afghanistan, aiding government forces fighting the Taliban and their supporters.