Police came under "heavy fire" during the attack last night in the eastern village of Awamiya, the birthplace of the cleric who was executed on Saturday, said the Saudi Press Agency.
The shooting left a civilian dead and wounded an eight-year-old child, said SPA, without specifying if any police were among the casualties.
Security forces were scouring the area for the assailants, the agency reported, citing a police spokesman in oil-rich Eastern Province.
The assaults on the diplomatic missions spurred the Sunni-ruled kingdom to sever ties with Shiite-dominated Iran, its long-time regional rival.
In recent years, the two countries have been divided over the war in Syria, where Iran is backing the regime, and the conflict in Yemen where a Saudi-led coalition is battling Shiite rebels.