Greater Hanish island was "cleansed in a well-executed operation conducted by members of the Popular Resistance supported by the joint coalition forces," said a statement on the official Saudi Press Agency.
The Popular Resistance is an umbrella of Yemeni fighters who have been battling Iran-backed Shiite rebels known as Huthis and are supported by coalition troops, air and naval forces.
Greater Hanish is part of an archipelago that commands access to the Bab al-Mandab Strait between the Red Sea and the Gulf of Aden, through which much of the world's maritime traffic passes.
Saudi Arabia's official Ekhbariya television broadcast images from the island of a damaged mosque and soldiers finding arms and ammunition.
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There were also images of warships and of helicopters hovering over the island.
"The operation comes simultaneously with ongoing operations by the Popular Resistance in the (northern) provinces of Hajja and Jawf to liberate them", the coalition said.
The rebels seized the Yemeni capital last year before advancing on other parts of the country.
The United Nations says more than 5,700 people have been killed, about half of them civilians, since fighting intensified in March.
A row between Yemen and Eritrea over the Hanish islands sparked an armed conflict between the two countries in 1995.
The dispute was resolved under a 1998 ruling by the International Court of Justice, which granted Yemen sovereignty over the volcanic islands.