"Today between 7am [0230 GMT] and 8am when she was [in her car] outside her house and going to her job, some gunmen attacked," Ayoobi said.
"Malalai Kakar died in front of her house. Her son was wounded."
Kandahar is a traditional stronghold of the Taliban, which is fighting an armed campaign against the Afghan government and US and Nato forces in the country.
Taliban claim
Kakar, who was in her late 30s, was shot in the head, a doctor said.
"She died on the spot and her son, who was badly injured, is in a coma in the hospital," he said without giving his name.
A spokesman for the Taliban said that the assassins were from his group.
"We killed Malalai Kakar," Yousuf Ahmadi, spokesman told the AFP news agency, said.
"She was our target, and we successfully eliminated our target."
Death threats
Kakar, a police captain and mother of six, led a 10-strong team of women police officers and had reportedly received numerous death threats in recent months.
She was well-respected in the police force for her bravery, one of her colleagues said on condition of anonymity.
Her work attracted attention from international media outlets interested in her work in Kandahar.
She was the first woman to enrol in the Kandahar police force after the 2001 removal of the Taliban and had been involved in investigating crimes against women and children, and conducting house searches.
Similar crimes
The head of Kandahar province's women's affairs department was killed in a similar way two years ago.
Assailants shot dead a female police officer in June in the western province of Herat in what was believed to be the first assassination of a female police officer in Afghanistan.
Afghanistan's police force was destroyed by the time the Taliban were removed and is being rebuilt with international assistance. It numbers about 80,000 people, including a few hundred women.
About 750 policemen have been killed in the past six months, mostly in Taliban-linked violence sweeping the country.