However, the governor office said nine civilians were killed in the attack. Another 89 were wounded.
"The fighting is over now". Some vehicles were burnt.
The 203rd Thunder Corps in a statement said the Afghan artillery units conducted strikes against Taliban positions in Nogha area of Ghazni city.
Eyewitnesses said following the first explosion, three back-to-back explosions were heard and the gunfight was still going on.
The talks have not stopped Taliban from launching attacks.
President Ashraf Ghani in a statement condemned the attack and said "terrorists" can not weaken the morale of the Afghan National Security and Defense Forces by such heinous attacks.
"Kandahar police headquarters initially came under a tactical bomb blast that enabled several martyrdom-seeking mujahedeen [holy warriors], equipped with heavy and light weapons, to enter the compound and launched [the] operation inside the [police] headquarters", the group asserted in a statement.
Pictures on social media showed a huge plume of smoke rising over Kandahar, and Hayat said about six trucks carrying commercial goods had burnt.
Kandahar is the former seat of the Taliban, who ruled Afghanistan from 1996 until their removal by a US-led coalition in 2001.
The news comes as 76 civilians were killed or wounded in separate attacks across the country earlier this week, despite the peace negotiations between the United States and the Taliban movement.
The Taliban also killed an Afghan commander on Wednesday in central Afghanistan.