The blast occurred in a western suburb of the city late on Sunday, said General Zalmay Hryakhail, a district police commander, on Monday. No one else was hurt.
The dead man's father and three brothers were arrested and police found another car already wired with explosives at their home, Hryakhail said.
He said the intended targets were not known, although he suspected that they may have been US forces or Nato-led peacekeepers who patrol the city.
The discovery came hours after Hamid Karzai, the Afghan president, said in an interview that he expected suicide attacks to continue "for a long time".
He said it was not clear who was behind them and believed that some of those responsible for about 20 attacks in the past four months may have been duped into killing themselves.
President Karzai had asked Mullah Mohammad Omar, the fugitive Taliban leader, to "get in touch" if he wanted peace, an offer that a Taliban official rejected on Monday.
Mullah Obaidullah Akhund, defence minister in the Taliban government ousted by US forces in 2001, called Karzai an "American puppet" who should be tried in an Islamic court.
"Hamid Karzai, the American agent, has turned Afghanistan into an American base and has killed thousands of Afghans," Akhund told Reuters by satellite phone from a secret location.
Karzai had told the Associated Press news agency on Sunday that he was willing to listen to what Omar had to say, but said that the Taliban leader would first have to account for his actions.
Karzai asked Mullah Omar 'to get
in touch' if he wanted peace
Akhund said it would be a betrayal of Islam to stop fighting "America and the infidel forces".
"Mullah Omar and his Taliban are not ready for this sin," Akhund said, adding suicide attacks would continue.
The Afghan president first held out an olive branch to the Taliban rank and file two years ago, and the head of a government commission for national reconciliation said in November talks were needed to end the violence that still plagues southern and eastern Afghanistan.
In other news, Abu Nasir al-Qahtani, an al-Qaida fighter who escaped from a US detention centre in Bagram, Afghanistan, has released a videotape.
"Hamid Karzai, the American agent, has turned Afghanistan into an American base and has killed thousands of Afghans"
Mullah Obaidullah Akhund,
Al-Qahtani, who escaped last year, said in the tape aired by Aljazeera on Monday that he led a December operation targeting a US military post in Khalbasat in eastern Afghanistan.
The videotape showed an attack by rocket-propelled grenades and light weapons as a US post resulting in vehicles being set ablaze.
Bodies of US soldiers could also been seen.
Aljazeera could not verify the authenticity of the videotape.