Daniel Rooney was 18 when he was shot in the St James Crescent of west Belfast in September 1972. He died later in hospital.
The circumstances leading to his death are disputed.
His body is being exhumed at Milltown cemetery, as part of an investigation into the Military Reaction Force (MFR).
The MRF was an Army unit operating in Belfast in the early 1970s.
In 2013, former members told the BBC's Panorama programme that as well as "hunting down" IRA members in Belfast, they shot and killed unarmed civilians.
Analysis - Panorama reporter John Ware
Panorama showed that, between May and September 1972, there were several unprovoked "drive-by" attacks on unarmed civilians by MRF soldiers similar to the one that killed Mr Rooney.
At the time, locals assumed the gunmen were loyalists. One attack is alleged to have been preceded by a friendly wave from the driver before a machine gun, protruding from a rear window, cut down three unarmed men chatting at a taxi rank.
On this occasion, the MRF soldiers were armed with a weapon, commonly used by the IRA, known as the "Chicago grinder".
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The investigation into the unit began two years ago, and encompasses 10 shootings.
It has been described as a "very large undertaking".
No post-mortem examination was conducted when Mr Rooney died in 1972, but it is now hoped that bullets could be found buried alongside him.
A team of experts including pathologists and forensic anthropologists has been involved in removing his remains.
Det Ch Insp Neil McGuinness said it was the most complicated case he has worked on during his 20-year policing career.
He confirmed it is a criminal investigation, and said anything found would be subject to forensic tests.
"Myself and my team are absolutely determined to get to the truth of what happened both for the Rooney family and all the other families involved in this investigation," he said.
That could mean interviewing soldiers based in Belfast at the time of Mr Rooney's death.
Although he described this as a massive investigation, Mr McGuinness said he does not expect there will be any other exhumations.
"This has been very extraordinary, very difficult for ourselves and the Rooney family.  It's challenging but it's necessary for the investigation."
Solicitor Paidraig O Muirigh, representing the Rooney family, said relatives were at the graveside.
He said: "The family are very emotional.  It's been distressing for them, and very challenging. But this is a necessity, they're supportive of the operation.
"The Rooney family hope that we can answer the question of whether there's further ballistic evidence available to assist the ongoing investigation and forthcoming inquest."
A medical report from Mr Rooney's entry to hospital confirmed there was an entry wound, but did not cite an exit wound.
Mr O Muirigh added: "That leaves the possibility that there is a bullet within the remains of Daniel Rooney."
Detectives from the Legacy Investigation Branch are conducting the operation at Milltown cemetery.
Two days have been set aside for the exhumation.
The 2013 Panorama programme was told the MRF consisted of about 40 men handpicked from across the Army.
Before it was disbanded 40 years ago, plain-clothed soldiers carried out round-the-clock patrols of west Belfast in unmarked cars.
One of them described their mission as "to draw out the IRA and to minimise their activities... if they needed shooting, they'd be shot".
Panorama identified 10 unarmed civilians - including Daniel Rooney - who were shot by the MRF, according to witnesses.
A senior police officer has said the exhumation of a man shot by the Army is the most complex investigation with which he has ever been involved.