The investigation by the Police Service of Northern Ireland (PSNI) follows a 2013 investigation by BBC Panorama into the Military Research Force, or MRF.
Daniel Rooney was 18 when he and a friend were shot by MRF soldiers firing from an unmarked car, as the youths stood on a street corner in the St James area of west Belfast.
The army claimed the men were armed. Eyewitnesses dispute this, consistent with forensic tests on Rooney and his friend that suggested neither youth was armed.
Mr Rooney was buried without a post-mortem examination. However, while a medical report recorded a bullet entry wound, there was no mention of an exit wound.
The PSNI want to see if the bullet can be retrieved from Mr Rooney's remains, assuming it was lodged in his body. It may then be possible to determine whether the bullet came from a military weapon.
The year of 1972 was the most violent of the 30-year Northern Ireland conflict. A total of 497 people died, mostly civilians killed in bombings or shot by the Provisional IRA and 151 soldiers and police officers were also killed.
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".
This was a Thompson sub-machine gun also favoured by the 1920s gangster Al Capone.
Panorama showed how in each disputed shooting, the MRF soldiers claimed they were fired on. Yet there was no independent evidence to show that any of the dead or wounded were armed, or that they provoked the attacks, or even that they were members of the IRA.
Two months after Daniel Rooney was shot, the military top brass decided the MRF was imperilling the army's already damaged reputation. Declassified papers show that a review of the MRF found there was "no provision for detailed command and control". The MRF was wound up after only 14 months.
When Prime Minister Edward Heath was told the MRF was to be replaced with the Special Reconnaissance Unit (SRU), he sent a message to the army emphasising that "special care should be taken" to ensure that the SRU should "operate within the law".
Det Ch Insp Neil McGuinness of the PSNI's Legacy Investigation Branch said: "Every effort has been made to complete" Rooney's exhumation "as sensitively and professionally as possible".
Last year the remains of a father of 12, Joseph Murphy, who was shot by the army in disputed circumstances, were exhumed.
Having been shot once, Mr Murphy was brought to an army base where he claimed a soldier shot him again in the place that he had been previously wounded. He later died in hospital. A bullet was recovered from his grave when his remains were exhumed.
The remains of a Belfast youth were being exhumed on Monday as part of a police investigation into his shooting by an experimental British army undercover unit in 1972.