The Ekhbariya state television station reported that the driver was a teenager on the run after killing his uncle. He tried to drive bypass the checkpoint, then triggered the blast after officers surrounded his vehicle south of the capital, Riyadh.
The use of an explosive so close to a prison holding hundreds of Islamist detainees will stoke fears of a growing threat of militancy in the world's top oil exporter.
State news agency SPA named the bomber as Abdullah Fahd Abdullah al-Rashed. It said he was born in 1997 and had never travelled outside of the kingdom.
SPA named the dead uncle as Rashid Ibrahim al Safyan, adding that Safyan had been a colonel, without elaborating.
The Islamic State group has called on supporters to carry out attacks in the kingdom and killed 25 people in two suicide bombings at Shi'ite Muslim mosques in the country's east in May.
"While security officers were manning one of the security checkpoints on Ha'er Road in Riyadh, they directed the driver of a suspected car to stop. The driver initiated an explosion which led to his death," the ministry said.
The road runs south from Riyadh to Ha'er prison, home to 1375 detainees mainly convicted of militant crimes.
The detention of thousands of Saudi Islamists accused or convicted of militancy over the past decade has angered many conservative Sunni Muslims in the kingdom, prompting some rare protests from 2011-13.
Two Saudis linked to a suicide bombing in Kuwait last month took part in those protests, local media reported at the time.
