According to the report, Abu Osama al-Muhajer, along with several other members of the organisation, were captured on 3 June.
Weapons, ammunition and telecommunication devices have been seized during the operation, the SPA reported.
In April, the terrorist group Daesh claimed responsibility for an attack on a Saudi police station northwest of Riyadh that left four militants killed.
Since 2015 Saudi Arabia, has commanded the Islamic Military Alliance to Fight Terrorism, an anti-terrorist coalition of Muslim nations tasked with defeating Daesh.
The Yemeni branch of Daesh conducted its first attack in March 2015, when it carried out suicide bombings on two mosques in Sana'a.
Since then, it has continued to carry out attacks aimed largely at civilian targets associated with the Shia Houthi movement.
*Daesh (also known as ISIS/ISIL/IS) is a terrorist group banned in Russia