Sgt Alexander Blackman, 42, from Taunton, Somerset, has served three years of a life sentence.
Five judges at the Court Martial Appeal Court in London ruled the conviction should be "manslaughter on the ground of diminished responsibility".
A further hearing will decide what sentence Blackman should serve.
Blackman - who was known as Marine A during the original trial process and fully identified when he was convicted - was not in court to hear the decision.
The marine had a recognised mental illness at the time of the shooting in September 2011, the hearing was told.
His defence team argued that fresh psychiatric evidence, if available at the time, would have provided him with the "partial defence of diminished responsibility".
The panel of judges were urged to overturn the "unsafe" murder conviction.
In Wednesday's ruling, the judges said:
Sgt Blackman's wife, Claire, was greeted by cheering supporters and honking taxi horns on the steps of the court before thanking the "tens of thousands of supporters, especially from the Royal Marines family, who have stood behind us throughout".
Mrs Blackman, who led a campaign alongside author Frederick Forsyth and the Daily Mail, said she was "delighted" with the ruling.
"This is a crucial decision and one which better reflects the circumstances my husband found himself in during that terrible tour in Afghanistan," she said.
Blackman's lawyer, Jonathan Goldberg, said Mrs Blackman and Mr Forsyth were the "the hero and the heroine of this occasion".
After the ruling, Mr Forsyth said: "It's not over yet. We always wanted justice - a very elusive word, much used, seldom achieved.
"It's a two-bladed weapon. Firstly, one blade to get a man who should never have been in prison out of prison, secondly, we go after those people who wrongly and, I think, villainously put him there."
By Jonathan Beale, BBC defence correspondent
Claire Blackman had a tear in her eye and could be heard saying "phew" when the Lord Chief Justice read out his judgement.
She was clearly relieved as Lord Thomas announced that her husband's murder conviction should be quashed and substituted with manslaughter.
When the news was relayed outside the court, there were celebrations.
Veterans, some wearing the green beret of the Royal Marines, cheered, prompted passing traffic to blast their horns.
This popular support has helped fund the appeal against his murder conviction.
Many view Blackman - better known as Marine A - as some kind of victim.
But he's not innocent in the eyes of the law. He's still guilty of manslaughter.
Blackman shot an insurgent who had been seriously injured in an attack by an Apache helicopter, in the chest at close range with a 9mm pistol.
Footage of the 2011 incident was captured on a camera mounted on the helmet of another Royal Marine.
A court martial heard that Blackman used abusive language and said: "There you are. Shuffle off this mortal coil."
He then turned to his comrades and said: "obviously this doesn't go anywhere, fellas. I just broke the Geneva Convention", it was alleged.
During his trial, Blackman, who denied murder, said he believed the victim was already dead and he was taking out his anger on a corpse.
He was convicted of murder in November 2013 and jailed for life.
He lost an appeal in May of the following year, but his 10-year minimum term was reduced to eight years.
The trial was the first time a member of the British armed forces had faced a murder charge in relation to the conflict in Afghanistan, which began in 2001.
A Ministry of Defence spokesman said: "We have fully co-operated with each stage of Sergeant Blackman's case and will continue to provide personal support to the family, as we have done since charges were first brought.
"We respect the court's decision and it would be inappropriate for us to comment further on it."
Panorama, Marine A: The Inside Story will be on BBC One at 22:50 GMT, and available later on iPlayer.
A Royal Marine jailed for shooting an injured Taliban fighter in Afghanistan has had his murder conviction reduced to manslaughter by an appeal court.