The attack Friday in the port city of Dammam and one a week earlier on another Shiite mosque in the nearby the village of al-Qudeeh have rattled the kingdom.
Both were claimed by the Islamic State militant group, raising concerns of further attacks in the OPEC powerhouse's oil-rich east.
The Interior Ministry identified the bomber in the most recent blast as 19-year-old Khalid Ayed Mohammed Wahabi Shammari.
The Islamic State group had earlier called the bomber a "soldier of the caliphate" going by Abu Jandal al-Jazrawi, a nom de guerre suggesting he was Saudi.
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The explosive used in that attack was RDX, a military-grade compound that was also deployed in the al-Qudeeh bombing.
That earlier attack killed 22 people, making it the deadliest militant assault in the kingdom in more than a decade.
The Islamic State group sees Shiites as apostates and seeks the overthrow of the Saudi monarchy. While Saudi authorities have promised to crack down on the extremists, many in its hard-line Wahhabist clerical establishment also consider Shiites to be heretics.
It promised a reward of 1 million riyals (USD 267,000) for information leading to the arrest of the suspects, 5 million riyals (USD 1.3 million) for the arrest of multiple suspects and 7 million riyals (USD 1.8 million) for thwarting a terrorist operation.