Courtney Harriot, Paul Green and Cleveland Davidson, all aged between 17 and 20 at the time, were arrested on the London Underground while travelling from Stockwell station, south London, in February 1972.
The trio, now in their late 60s, and three friends - who became known as the "Stockwell Six" - were put on trial at the Old Bailey, largely on the word of British Transport Police officer Detective Sergeant Derek Ridgewell.
The Stockwell Six were accused of trying to rob Ridgewell, who was in plain clothes and had previously served in the South Rhodesian, now Zimbabwean, police force.
They all pleaded not guilty, but all bar one were convicted and sent to jail or Borstal, despite telling jurors that police officers had lied and subjected them to violence and threats.
Their convictions were referred to the Court of Appeal by the Criminal Cases Review Commission on the basis that there is "a real possibility that the Court of Appeal will now quash those convictions".
At a hearing at the Royal Courts of Justice in London on Tuesday, the Court of Appeal cleared Mr Harriot, Mr Green and Mr Davidson, nearly five decades after they were convicted.
Sir Julian Flaux, sitting with Mr Justice Linden and Mr Justice Wall, said: "It is most unfortunate that it has taken nearly 50 years to rectify the injustice suffered by these appellants."
The judge added: "These appeals are allowed and the convictions are quashed."
The two remaining members of the Stockwell Six who were convicted have not been traced.
Previously, the Criminal Cases Review Commission said it is still very keen to hear from the remaining members.
In December 2019, three members of the "Oval Four" - who were arrested at Oval Underground station in 1972 and accused of stealing handbags by Ridgewell's "mugging squad" - also had their convictions overturned.
Officer Ridgewell was involved in a number of high-profile and controversial cases in the early 1970s, culminating in the 1973 acquittals of the "Tottenham Court Road Two" - two young Jesuits studying at Oxford University.