The Royal Marine, known only as Marine A, was found guilty of murdering the unknown Afghan national in the southern Helmand Province in 2011.
Two others, Marines B and C, were acquitted of the charges against them at the court-martial in Bulford, southwest England.
As the victim convulsed and died in front of him, Marine A quoted a William Shakespeare phrase.
"There you are. Shuffle off this mortal coil, you xxxx. It's nothing you wouldn't do to us," Marine A told the insurgent after shooting the man in the chest from close range with a nine-millimetre pistol.
He then turned to his comrades and said: "Obviously this doesn't go anywhere, fellas. I just broke the Geneva Convention."
The execution was filmed by a camera on Marine B's helmet.
Marine A was convicted by a seven-strong board, consisting of officers and non-commissioned officers, following a two-week trial.
He faces life imprisonment. He will be sentenced on December 6, when the court will announce the minimum term he will serve.
Marine A had denied murdering the captured man.
Marines B and C were acquitted of having been "party to the killing" and having "encouraged and assisted" their comrade in committing the murder.
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