Polls closed at 5 p.m. local time but election officials could not immediately determine the turnout from the 567,000 registered voters in Kandahar.
But some polling stations and centers had still not opened by 8am - despite dozens of people having lined up from early Saturday morning.
Afghanistan's first parliamentary elections since 2010 were held against a backdrop of near-daily attacks by the Taleban, who have seized almost half the country and have refused offers to negotiate with the government.
The vote in Kandahar had been delayed by a week after the murder, on October 18, the controversial but respected provincial chief of police, general Abdul Raziq, a man of a strong anti-taliban regarded him as a bulwark against the insurgency in the south.
Bashir Ahmad, another resident of Kandahar, who is disabled and came to cast his vote, called voting vital for the future of Afghanistan and criticized the IEC for not starting the process on time.
Any citizen suspecting or witnessing fraud or irregularities should channel complaints through the relevant Afghan authorities, especially the Electoral Complaints Commission.
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Miller escaped unhurt, but US Brigadier General Jeffrey Smiley was among 13 people wounded in the shooting claimed by the Taliban. Yet millions of Afghans have defied Taliban threats and waited, often for hours, to cast their votes.
On the eve of the ballot, Afghan air strikes killed at least 56 Taliban militants in Kandahar's Shah Wali Kot district, provincial police spokesman Zia Durrani told AFP.
According to it, almost four million people voted last weekend, in spite of many attacks perpetrated against the voters.
IEC figures show roughly 4.2 million out of the almost nine million people registered to vote actually cast a ballot.
The first parliamentary elections since 2010 are being held against a backdrop of near-daily attacks by the Taliban, who have seized almost half the country and have repeatedly refused offers to negotiate with the Afghanistan government.
These elections have been seen as a test of the state of Afghanistan's fragile democracy as well as a trial ahead of the presidential elections set to take place in April 2019.