Women and children were among those injured in the attack at the gate of the complex in Zabul province, which includes several government buildings including the provincial council.
The attack comes as the Taliban step up attacks on government and foreign targets despite Kabul's repeated overtures to the insurgent group to reopen peace negotiations.
"Around 40 people were wounded in the suicide attack on the complex" in the provincial capital Qalat, local police chief Mirwais Noorzai told AFP.
"Around 1,000 kilograms of explosives were used in the attack," he added.
Deputy police chief Ghulam Jailani Farahi confirmed the casualties, adding that the wounded were all civilians and some were in critical condition.
But provincial council chief Atta Jan Haqbayan gave a higher toll, saying 67 people were wounded, including 17 women.
The Taliban, waging a 13-year war against the US-backed Afghan government, claimed responsibility for the attack.
"As part of the Azm operation, this afternoon a martyrdom seeker... conducted an attack on the provincial council, where cruel and unjust decisions against Muslims and Islam were being taken," Taliban spokesman Qari Yousuf Ahmadi said in a statement.
The Afghan Taliban launched their annual spring-summer offensive -- titled 'Azm' (Determination) -- in late April, vowing nationwide attacks in what is expected to be the bloodiest fighting season in a decade
The insurgents have launched a series of attacks in the capital and around the country as NATO forces pull back from the frontlines.
A blast triggered by a Taliban car bomber ripped through the parking lot of the justice ministry in Kabul on May 19, killing four people and wounding dozens of others.
Also this month, 14 people -- mostly foreigners -- were killed in a Taliban attack on a Kabul guest house that trapped dozens attending a concert.
Official efforts to bring the Taliban, who have waged a 13-year war to topple the US-backed government, to the negotiating table have so far borne little fruit.
The surge in attacks has taken a heavy toll on civilians, according to the United Nations Assistance Mission to Afghanistan.
In the first four months of 2015, civilian casualties jumped 16 percent over the same period last year, it said.
The Afghan government has drawn public criticism for its inability to end insurgent attacks -- a fact partly attributed by critics to political infighting and a lengthy delay in finalising a cabinet.
President Ashraf Ghani on Thursday nominated Mohammad Masoom Stanekzai, a top official in the government body overseeing the country's peace process, for the crucial position of defence minister.
The post had been left vacant for months due to disagreements between Ghani and his chief executive officer and former presidential election rival, Abdullah Abdullah.
Public criticism over the failure to appoint a defence minister has been especially fierce.
Afghan forces are now solely responsible for security after NATO's combat mission formally ended in December, with a small follow-up force staying on to train and support local personnel.
Earlier this month NATO formally announced plans to retain a small military presence in Afghanistan after 2016 to help strengthen local security forces.
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