Some reports said there were two suicide bombings.
The Taliban claimed responsibility for Thursday's attack.
Among those killed was the provincial police chief, Torialai Abdyani.
"At around 2.50 a.m.in the morning (2220GMT Wednesday), a group of attackers attacked the Afghan National Army base in Maiwand district of Kandahar. The terror attacks demonstrate that safe havens and support systems continue to be available to the terrorists", the ministry's statement further said. Other militants pushed in and started a firefight with army soldiers.
Thursday's attacks take the number of suicide and gun assaults on security installations this week to four and increases the total death toll to more than 120, including soldiers, police and civilians.
The base itself was left in ruins, officials said.
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Taliban attacks on Afghan military and police compounds have increased this week after USA officials met in Oman with delegations from Afghanistan, Pakistan and China to discuss the possibility of restarting peace negotiations with the insurgent group.
US President Donald Trump vowed earlier this year to stay the course in Afghanistan, America's longest war.
They began hours after United States drone strikes in the tribal region along the border with Pakistan killed more than two dozen militants, according to sources and local officials.
After the attack, Afghan security forces said that they had launched air and ground operations in 17 provinces that had killed 102 insurgents. The battle ended with a USA airstrike that killed nine Taliban fighters, North Atlantic Treaty Organisation officials said.
In the western province of Farah, the authorities said that militants attacked a government compound in the Shibkho district, killing at least three police officers.
Defence ministry spokesman Dawlat Waziri told AFP news agency the assailants detonated at least one explosive-packed truck and razed the compound.
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