No one has claimed of responsibility for the attack.
Asked if some of those arrested had been charged for the killing, Turki said: "Not yet, we still have to complete the investigation".
Security forces are looking for two more Saudi nationals, who were named as Abdallah Sayer al-Mohammadi and Nasser bin-Latif al-Balawi, the interior ministry said in a statement.
"We just want them to appear at the nearest police station and clear their names," Turki said.
The ministry gave the two men 24 hours to turn themselves in and offered $1.9 million for information leading to their arrest.
Saudi Arabia has vowed to crack down on Islamic militants and has condemned the attack.
Fighters swearing allegiance to al-Qaeda launched a campaign to topple the US-allied Saudi monarchy in 2003, carrying out attacks on foreigners and government installations, including the oil industry.
Tough security measures and a powerful publicity campaign helped crush the violence but analysts and diplomats have said the underlying drives of radical Islamic ideology and anger at Western policy in the region remain strong.