Militant gunman then opened fire from nearby positions and members of the security forces were battling them, according to Tadeen Khan, the southern city's chief of police.
Hayatullah Hayat, the provincial governor for Kandahar, said Afghan security forces had launched a "clean-up operation" to see if any attackers were remaining.
Eyewitnesses said that following the first explosion, three back-to-back explosions were heard and the gunfight was still going on.
Qari Yusuf Ahmadi, a Taliban leader, said in a statement that the attack was carried out by suicide bombers "armed with light and heavy weapons and explosives vests". Some vehicles were burnt.
Pictures on social media showed a huge plume of smoke rising over the troubled city.
The Taliban claimed responsibility for the attack.
The Taliban say they will hold talks with representatives of a Swedish non-profit group after threats by the insurgents forced the organization to close 42 clinics it runs in eastern Afghanistan.
Details have emerged on an insider attack on Wednesday in Afghanistan, when an Afghan soldier shot and killed the commander of the Afghan Army's 3rd division, Col. Mateen Mujtaba, in the Ghazni Province.
Resolute Support, the US-led North Atlantic Treaty Organisation mission in Afghanistan, said no USA or coalition troops had been killed or wounded, but declined to comment further. He escaped unhurt, but US Brigadier General Jeffrey Smiley was among 13 people wounded in the shooting, which the Taliban said had targeted Miller and Raziq.