The attack took place close to the area used by military vehicles and targeted a European police training mission vehicle.
Two Afghan teenage girls, described as bystanders, died in the blast, while at least 18 people were injured.
The Taliban said it carried out the attack in a statement emailed to media.
The British victim has not been named but UK Foreign Secretary Philip Hammond said he was a security contractor.
Mr Hammond said he "strongly condemned (the Taliban's) cowardly actions".
A spokeswoman for the EU's police mission (Eupol), Sari Haukka-Konu, said one non-mission member who had been travelling in the Eupol vehicle had been killed.
The attack is the second major incident in Kabul within a week. A Taliban gun attack on a hotel in the city on Thursday killed 14 people, most of them foreigners.
At least eight women and three children were among the wounded in Sunday's attack, Kabul police said, in addition to three members of Eupol.
The powerful bomb sent up a plume of smoke, which cleared to show a scene of carnage, the BBC's David Loyn in Kabul reports.
Rescuers struggled to carry wounded civilians from the scene in the confused aftermath of the bombing, and a doctor at a nearby hospital dealing with the wounded was reported to have found some of his own relatives among them.
A spokesman for the interior ministry, Najib Danish, told reporters that one foreign military vehicle and two civilian vehicles were damaged in the attack.
The bombing on Sunday comes two weeks after the first exploratory peace talks between the Taliban and a group of negotiators including representatives of the government.
Our correspondent says that it is clear that while talking peace, some elements in the Taliban are keeping up the pressure.
A suicide bomb attack near the entrance of the international airport in the Afghan capital Kabul has killed three people, including a British citizen.