According to police, a suicide bomber on foot detonated his explosives close to the gate of the offices.
The attacker was "identified and gunned down by police before reaching his target", Kabul police spokesman Basir Mujahid told reporters.
It was not immediately clear how many people were at the site at the time of the attack.
But long queues of voters had formed as early as 07: 00 local time (02h30 GMT), the polls opened two hours late.
"People are determined to take part in the election because they want to show the Taliban that terror can not stop the Afghans from deciding their future", Dawa Khan Menapal, an independent candidate, said before the voting in Kandahar.
Figures by the electoral body indicate that around four million people risked voting in the parliamentary elections, which the Taliban militant group had vowed to attack.
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The policeman, who was martyred, was injured in the attack and later died of his wounds.
According to the election commission, 567,000 people have registered to vote in Kandahar. But hiccups with biometric devices and voter lists persisted.
The election is seen as a dry run for next year's presidential vote and an important milestone in advance of a United Nations meeting in Geneva in November where Afghanistan is under pressure to show progress on "democratic processes".
An independent monitoring group said voting also was marred by ballot stuffing and intimidation by armed men affiliated with candidates in 19 of the country's 32 provinces that were voting last weekend.
Preliminary results of nationwide voting are not expected before mid-November.
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.