A military statement said: "When the force entered the target buildings, they encountered two armed men. Perceiving hostile intent, coalition forces engaged and killed them.
"It was subsequently determined that the two armed men killed in the incident were related to the governor of Salaheddin province."
'Barbaric and inhuman'
Lieutenant-Colonel Saad al-Qaisi, al-Shakti's brother, said American troops stormed a family house in the town of Beiji, where the governor's son Husam and his cousin were staying.
He said: "They shot dead Husam and wounded three others. This is barbaric and inhuman."
Al-Shakti cut short a visit to Turkey because of the shooting, local officials said.
Abdullah Jabara, the deputy governor, said: "We demand an investigation into this incident."