The interior ministry named the bomber as Fahd Suliman Abdul-Muhsen al-Qabaa and said he flew into Kuwait's airport at dawn on Friday, only hours before he detonated explosives at Kuwait City's Imam al-Sadeq mosque.
Militant group Islamic State claimed responsibility for the suicide bombing at the mosque, where 2,000 worshippers were praying at the time. It was one of three attacks on three continents that day apparently linked to hard-line Islamists.
"The interior ministry will continue its efforts to uncover the circumstances of this explosion," interior ministry spokesman Adel Hashash told Kuwait state television.
The bombing has sharply heightened regional security concerns because Islamic State appears to be making good on its threat to step up attacks in the holy fasting month of Ramadan.
The group, seeking to expand from strongholds in Iraq and Syria, says its priority target is the Arabian Peninsula and in particular Saudi Arabia, home of Islam's holiest places, from where it plans to expel Shi'ite Muslims.
The Islamic State subscribes to a puritanical school of Sunni Islam that considers Shi'ites as heretics.
The ministry said the driver of the Japanese-made car, who left the mosque immediately after Friday's bombing, was an illegal resident named Abdul-Rahman Sabah Aidan.
The interior ministry, which had earlier reported the vehicle owner's arrest, said Aidan, 26, was found hiding in one of the houses in al-Riqqa residential area.
"Initial investigations showed that the owner of the house is a supporter of the deviant ideology," the ministry said, employing a term often used by authorities in the Gulf Arab region to refer to hard-line Islamist militants.
The owner of the house, a Kuwaiti citizen, was also detained, the ministry said.
Officials said the bombing was clearly meant to stir enmity between majority Sunnis and minority Shi'ites and harm the comparatively harmonious ties between the sects in Kuwait.
Shi'ites are between 15 and 30 percent of the population of Kuwait, a mostly Sunni country where members of both communities live side by side with little apparent friction.
Kuwait is a conservative Muslim country where alcohol is banned, but it is less strict than Saudi Arabia on issues such as women's rights and freedom of religion.
Kuwaitis reacted with outrage to the bombing. Some said citizens who fund Islamist armed groups fighting in Syria and Iraq were to blame for any militancy in Kuwait.
"The wrath of God will come upon ISIS and everyone who is supporting them and collecting funds for them under the cover of helping refugees and orphans," wrote Hamad al-Baghli, a Kuwaiti, on Twitter.
"Everyone who is funding and donating to ISIS should be charged with treason because they want to burn Kuwait," tweeted Asmaa Asiri.
In other attacks on Friday, a gunman in Tunisia killed 37 people including Western tourists on a beach, and in France a decapitated body was found after an attacker rammed his car into a gas container, triggering an explosion.