Keith Maupin, 20, of Batavia, Ohio, who was listed as captured on 16 April by the US Department of Defence (DOD), was executed because the Bush administration did not change its policies in Iraq, the statement said.
The group - calling itself the Persistent Power Against the Enemies of God and the Prophet - claimed the execution was also in revenge for what it called atrocities committed against "Muslim martyrs" in Iraq, Saudi Arabia and Algeria.
In the videotape, which was aired on the Aljazeera early on Tuesday, Maupin is seen talking to the camera.
He is dressed in military fatigues.
Aljazeera said it catered to viewers' sensitivities and did not broadcast the execution but said Maupin appeared blindfolded in front of a freshly-dug grave with his back to the camera.
He was then shot in the head.
According to a DOD statement, Maupin and Sergeant Elmer Krause, 40, of Greensboro, North Carolina, were categorised as duty status - whereabouts unknown on 9 April.
Individuals using rocket-propelled grenades and small arms fire attacked their convoy. Both soldiers were assigned to the Army Reserve's 724th Transportation Company, from Bartonville, Illinois, the statement said.