Police said the suicide bomber was on foot and detonated the explosives near the gate of the offices.
Elections in Kandahar was delayed after the provincial police chief Gen. Abdul Raziq was assassinated in a shootout just two days ahead of the election day.
On October 23, Taliban forces abducted 20 Afghans in Farah Province because of participating in the country's parliamentary elections and killed two of them.
More than 4 million people cast their ballots elsewhere last weekend despite violence and logistical issues, the Independent Election Commission (IEC) said.
It was not clear how many people were present at the time of the attack around the election commission's sprawling compound located near an arterial road in the city.
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.
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The policeman, who was martyred, was injured in the attack and later died of his wounds.
More than half a million people were registered to vote across the province, according to the IEC's media relations office, with women making up roughly 1 in 8 registered voters.Turnout figures have not yet been released.
These legislative elections are the first in Afghanistan to make use of biometric devices at voting centers.
On Saturday voters in Kandahar - the southern birthplace of the Taliban and a province notorious for ballot stuffing - went to the polls.
Preliminary results of nationwide voting for Afghanistan's 249-seat Parliament are not expected before mid-November.
So far, the electoral body has received thousands of complaints following the vote.
Kandahar is the birthplace of the Taliban and has always been a center of the insurgency that has been waged throughout years of worldwide intervention since the Taliban were toppled during a United States bombing campaign in 2001.