They said the apparent US strike on their car was carried out late Thursday in the Sumaa area of the province.
They said it left the terrorist's body "incinerated" and his two bodyguards dead.
A similar drone strike on Sunday in Marib province, east of the Yemeni capital Sanaa, killed five suspected Al-Qaeda terrorists.
The United States, the only government to operate drones over the impoverished country, considers AQAP to be the terrorist group's most dangerous branch.
Washington has intensified its drone war against Yemen-based terrorists since US President Donald Trump took power in January.
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