"A wanted man was killed. He was a Saudi," Kuwait's interior minister, Shaikh Nawaf al-Ahmad al-Sabah, told reporters outside a house stormed by security forces in Umm al-Haiman, south of Kuwait City.
It was the first time a Saudi was identified as having been involved in such an incident in Kuwait.
"We arrested another and the rest fled," the minister said after the clashes, which erupted when security forces hunted down suspected fighters.
"The man killed carried a Saudi passport, while the one arrested is a Kuwaiti citizen," Shaikh Nawaf said.
"Those who fled do not number more than six," the minister said.
The incident follows a clash last Monday in a Kuwait city suburb between Islamists and state security forces in which two members of state security and one gunman were killed.
The sources said Saturday's firefight took place in a residential neighbourhood of Umm al-Haiman, in Ahmadi governorate which houses Kuwait's largest refinery, the 460,000 barrel per day Mina al-Ahmadi coastal plant.
But they said there was no indication that the attack had been aimed at any of the country's oil facilities.
The sources said the interior ministry had put on alert all security personnel across the tiny oil-rich nation, which controls about 10% of global petroleum reserves.
Kuwaiti authorities have been cracking down on Islamist fighters since Monday's fatal shootout and have so far rounded up at least 40 suspected fighters for questioning.
Earlier on Saturday, Energy Minister Shaikh Ahmad al-Fahd al-Sabah said the security forces had to sort out the problem of Islamist fighters.
"The security apparatus must finish this issue at this stage"
Energy Minister Shaikh Ahmad al-Fahd al-Sabah
"The security apparatus must finish this issue at this stage and we must not shy from coordinating with our neighbours in exchanging information so that we can surround this group and control it," the Kuwaiti minister said.
Shaikh Ahmad also urged Muslim clerics "to take control of them (militants) ideologically so that there is a correct treatment for such an issue".
Kuwait, where about 25,000 American soldiers are based, has been cracking down on activists opposed to the presence of US forces in the country.