Security sources said one of the suspects, a wanted Kuwaiti man identified as Fawaz al-Utaibi, died later in hospital of his wounds. Two state security officers were wounded in the shootout that took place on Monday.
Shortly before the incident, the US embassy in Kuwait warned that it had information of imminent drive-by attacks on Westerners in the US-allied Gulf Arab state.
In an advisory, the US embassy said it was issuing "this urgent message because it has received credible information that an individual or individuals moving about Kuwait in a black-coloured small sedan intend to randomly attack Westerners".
A US embassy security official at the scene of the shootout said the car was the one used by al-Utaibi and his accomplices.
Security men killed
The Interior Ministry said a first lieutenant and a sergeant had been killed and two other security personnel wounded in the exchange of fire with al-Utaibi as they pursued him in the Hawalli suburb of Kuwait City.
It said al-Utaibi had entered a shop in Hawalli and opened fire at security personnel who followed him inside.
Al-Utaibi then ran outside and got into a car where accomplices were waiting. One of the passengers opened fire on the security officers, killing two and wounding two, the ministry added.
The attack took place a day after
Kuwait lifted a security alert
Security sources said al-Utaibi had been wounded in the shootout and taken to hospital where he later died.
The sources said al-Utaibi had been returning a car to a rental company when state security personnel decided to arrest him. At least one accomplice in the car opened fire when he saw al-Utaibi being pursued, one source said.
The Hawalli incident happened a day after Kuwait lifted a high security alert imposed two weeks ago. Under the measures, security personnel in armoured vehicles fanned out in key areas, especially around vital oil installations and western embassies.