Ibrahim al-Rubaish, who in January accused France of surpassing the United States as the top enemy of Islam, was killed with several other militants in a "crusader raid" on Monday, the statement said.
The statement published online was apparently referring to a drone attack that killed six suspected jihadists in their vehicle in Hadramawt province in southeast Yemen.
The United States, whose intelligence agencies consider AQAP to be the most dangerous branch of the jihadist network, is the only country that operates drones over Yemen.
He fought in Afghanistan and was held in the US military prison in Guantanamo for "a few years" before he joined AQAP following his release, it said.
Al-Qaeda militants have exploited the breakdown of security as a Saudi-led coalition last month launched an air campaign against rebels amid fierce fighting between rival Yemen forces.
AQAP has seized Mukalla, the capital of Hadramawt province.
US Defence Secretary Ashton Carter acknowledged last week that Al-Qaeda was making gains and that the fighting in Yemen was complicating counter-terrorism efforts, but vowed that they would continue.
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