Parwiz Najib, the director of the Kandahar governor's media office, said two of the governor's bodyguards were killed and one wounded in an exchange of gunfire with the attackers, who were equipped with explosive-rigged suicide vests, hand grenades and automatic rifles, AP reported.
The attackers were attempting to reach the governor's office inside the compound when the governor's bodyguards identified them, sparking the gunbattle, Najib said.
Najib said the attack lasted less than 30 minutes and the governor was unharmed.
Taliban spokesman Qari Yousef Ahmadi claimed responsibility for the attack.
Saturday's firefight is another reminder of the insurgents' ability to strike in even some of the most secure areas of the country. Earlier this month, militants launched a large-scale coordinated attack on the diplomatic center of the capital Kabul and three other cities in which 36 insurgents and 11 others were reported killed.