Shortly afterward, a second blast went off higher up on the same street.
The bomber was killed and two others were injured. Witnesses reported seeing a man lying on the ground afterward with blood appearing to come from his abdomen.
The exploded car contained gas canisters, rescue workers said.
Police aren't commenting on a motive or confirming that it was a suicide attack.
It would be the first such attack in the Nordic country.
Swedish Foreign Minister Carl Bildt described the attack as "a most worrying attempt at a terrorist attack." Bildt commented in a Twitter message that it "failed - but could have been truly catastrophic."
A Swedish news agency aid it received an e-mail threat just before the blast in which the writer claimed to have visited the Middle East "for jihad," and referred to the country's soldiers in Afghanistan and a cartoon of the Prophet Muhammad that outraged the Muslim world.
Police said they were aware of the e-mail, which had also been addressed to Sweden's security police, but couldn't immediately confirm a link to the explosions.
The Associated Press contributed to this report.
