The attack on one of the largest air bases in Afghanistan coincided with a regional peace conference in Pakistan where President Ashraf Ghani asked Pakistan to help restart peace talks with the Taliban that were broken off earlier this year.
The ministry said operations against the attackers, who took up position in a part of the complex used for accommodation, had proceeded slowly to limit the number of civilian casualties.
In all, 38 civilians, 10 soldiers and two police officers were killed in the fighting, while 37 security forces and civilians were wounded.
No casualties were reported among the thousands of international military and contractors based at the heavily fortified site.
Peace conference
The raid in one of the Taliban's traditional strongholds coincided with the start of a regional security conference in Islamabad, where Afghan President Ashraf Ghani made a plea for more support from neighbors to fight the growing insurgency.
The Taliban, fighting to re-establish hard-line Islamist rule after U.S.-led military intervention ousted them from power in 2001, have been struggling to settle a leadership dispute which has seen scores killed in fighting between factions.
But the attack in Kandahar showed their continued ability to inflict damage and will also increase pressure on Ghani's government to contain the spreading insurgency, which has claimed thousands of civilian and military casualties since the start of the year.
Officials said fighters attacked a perimeter area of the huge and heavily fortified complex on Tuesday evening, initially taking up position in a school in a residential area of the site, which houses both a civilian airport and military base.
NATO military personnel, civilian contractors and Afghan forces are based there.
Earlier, the Taliban said in a statement 150 soldiers had been killed after suicide attackers had entered the base and attacked international forces and their Afghan allies. But the militants have often made exaggerated casualty claims in the past.
The attack follows an earlier incident in which two suicide bombers died attacking a police station in Kandahar.
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