The blast was the latest incident in Qatif, which has seen criminal violence as well as protests among the Sunni- dominated kingdom's Shiite minority.
Police spotted a stolen sports utility vehicle used "to carry out terrorist and criminal crimes" and then took "necessary action", a ministry statement said, without detailing what tactics police used.
"As a result, it caught fire and exploded," killing its two occupants.
"An investigation is underway to reveal the identity of those two individuals."
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Footage that circulated on social media showed thick smoke rising from a vehicle engulfed in flames, with periodic sounds of explosions.
Saudi Arabia's Eastern Province, where most of its Shiite minority live, has been rocked by unrest since 2011.
The authorities refer to the protesters as "terrorists" but also use the term to refer to those engaged in criminal activity and to Sunni extremists of the Islamic State group, who have carried out a number of attacks on Shiites.
Last month, violence escalated around a redevelopment project in the old Almosara section of Awamiya, a Qatif district town.
Three United Nations rights experts called on Saudi Arabia to halt demolition of the 400-year-old Almosara neighbourhood which, they said, reportedly caused "injury, deaths and material losses to the civilians".
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