British embassy staff were also injured in the attack, though no further details were immediately released. Among the bystanders injured were five children, officials said.
The huge explosion could be heard across Kabul and a plume of smoke rose above the attack site on the Jalalabad road, a main route that houses many foreign compounds and military facilities.
The attack, coming three days after two US soldiers were killed in another bomb blast in Kabul.
The Taliban said they were behind Thursday's attack in a message on a recognised Twitter account, claiming that many foreigners had been killed.
39 Killed and Injured in a Blast Targeted British Embassy Vehicle
British ambassador Richard Stagg was at a televised meeting and not in the vehicle when it was hit.
"We can confirm that a British embassy vehicle has been attacked in Kabul. A number of staff have been injured," a British foreign office spokeswoman said.
Afghan officials at first reported the bomber was on a motorbike, but later said he was driving a Toyota Corolla car.
Kabul blasts
Kenishka Turkistani, spokesmen for the ministry of public health, said five bystanders were killed and 34 wounded in the attack.
"They are all Afghan civilians, the wounded include five children. We don't record foreign casualties," Turkistani said.
War-torn Afghanistan suffered its deadliest attack of 2014 on Sunday when a suicide bomber struck at a volleyball match in the eastern province of Paktika, killing 57 people.
At least seven blasts have hit Kabul over the last ten days, including attacks on foreign compounds and on a female Afghan MP who was injured in a suicide bombing that targeted her car.
The US-led NATO combat mission in Afghanistan will finish at the end of this year, with about 12,500 troops staying on into 2015 to train and support the Afghan army and police.
President Ashraf Ghani, who came to power in September, has vowed to bring peace to Afghanistan after decades of conflict, saying he is open to talks with the Taliban, who ruled Kabul from 1996 to 2001.
Afghan investigators inspect wreckage at the site where a suicide bomber targeted a vehicle convoy of Afghan lawmakers in Kabul on November 16, 2014. A prominent female MP and 17 others injured and 3 killed in this blast, in which the attacker detonated an explosives-packed car, left the MPs' vehicles badly damaged on a main road in the west of Kabul, close to the parliament.
Ghani finally emerged as president after signing a power-sharing deal with his poll rival Abdullah Abdullah.
Both men claimed to have won fraud-tainted elections in a prolonged stand-off that caused political paralysis in Kabul and fuelled worsening violence nationwide.
Britain this week ended its 13-year military presence in southern Afghanistan when the last Royal Air Force personnel departed Kandahar airfield.
The British military contribution next year will be the supervision of an officers' training academy outside Kabul.
But US President Barack Obama has quietly approved guidelines to allow the Pentagon to target Taliban fighters in Afghanistan, Broadening previous plans that had limited the military to counterterrorism missions against al-Qaida after this year, U.S. officials said.