No one immediately claimed responsibility for the assault - the third major attack on NATO forces in Afghanistan in six days - but the Kandahar area is a Taliban stronghold.
On Tuesday, a Taliban suicide bomber attacked a NATO convoy in the capital, killing 18 people including six NATO service members including five Americans and a Canadian.
The next day, dozens of Taliban militants attacked the main U.S. military base - Bagram Air Field - killing an American contractor in fighting that lasted more than eight hours.
Navy Cmdr. Amanda Peterseim, a spokeswoman for NATO forces at Kandahar, said at least five rockets struck the base in the initial attack, but witnesses said explosions continued through much of the night. There were no reports of deaths and she did not have the precise number of wounded.
"The alarm has been sounding for several hours, but no insurgents have penetrated the base perimeter," NATO said in a statement. It said "a number" of military and civilian personnel were wounded "and are receiving medical treatment. There are no confirmed fatalities."
NATO said troops and civilians were told to remain in bunkers as a precaution.
Peterseim did not know how many insurgents launched the attack but said they did not appear to be wearing suicide vests, as had many of those who stormed the Bagram Air Field north of Kabul on Wednesday. In addition to the U.S. contractor's death, 16 militants were killed and five attackers were captured in the Bagram assault.
Rocket attacks against the Kandahar base, located about 10 miles (16 kilometers) south of Kandahar city, are not uncommon. But ground assaults against such large facilities as Kandahar and Bagram are rare. Two attacks in the same week show that the militants are capable of complex operations despite NATO military pressure.
The attacks follow a Taliban announcement of a spring offensive against NATO forces and Afghan government troops - their response to a promise by the Obama administration to squeeze the Taliban out of their strongholds in southern Kandahar province.
Kandahar Air Field is the launching pad for thousands of additional U.S. forces pouring into the country for a summer surge against the Taliban.
Attacks in the south earlier Saturday killed three NATO service members - one American, one French and one Dutch - and an Afghan interpreter. That brought to 997 the number of U.S. service members who have died since the war began in October 2001, according to an Associated Press count. The Dutch death toll in Afghanistan now is 24 and the French toll is 42.
A loudspeaker announcement at the Kandahar base said the ground attack was coming from the north, said Maura Axelrod, a reporter with HDNet who was inside the base. She said she could hear heavy outgoing fire and that commanders had come into the bunker where she had taken cover to order all Marines with weapons to help in establishing a security perimeter.
An Afghan named Najibullah who works with a private security company on the base said that he heard rockets hitting for about half an hour. He only gave one name.
NATO's current push is aimed at winning over the population in Taliban-friendly areas by establishing security and bolstering the local government. However, each military strike has created potential for backlash amid arguments about who is truly an insurgent.
In the latest such incident, at least a dozen people were killed south of the capital Saturday after U.S. troops spotted two insurgents trying to plant bombs, an Afghan official said.
The two were shot dead in Paktia province, district chief Gulab Shah said. Troops saw comrades drag the two bodies away and called in a helicopter gunship which killed 10 more people, whom U.S. officials said were all militants, Shah said.
Shah said Afghan authorities will investigate to make sure the dead were all insurgents.
Civilian deaths are a flashpoint issue in Afghanistan, where President Hamid Karzai has urged NATO to take all necessary measures to protect civilian lives.
More than eight years into the war in Afghanistan, international support is also weakening.
The defense minister of Britain's new Conservative-Liberal Democrat coalition government met with Karzai in Kabul on Saturday and said he hopes to speed the withdrawal of British troops.
Defense Secretary Liam Fox is quoted in Saturday's edition of The Times newspaper he "would like the forces to come back as soon as possible," and wants to see if it is possible to speed the training of Afghan troops.