Mr Conlon died last weekend at his Falls Road home in west Belfast with his sisters by his side after a period of illness. He was aged 60.
On Saturday, family and friends turned out at St Peter's Cathedral for his Requiem Mass.
The funeral cortege left Mr Conlon's sister Ann's home on the Falls Road - she was pictured with Mr Conlon in 1989 when he was freed at the Old Bailey.
Mr Conlon and three others - Paddy Armstrong, Paul Hill and Carole Richardson - known as the Guildford Four, were wrongly given life sentences for the 1974 attack which killed five people and injured 65.
At the time of their sentencing, the trial judge Mr Justice Donaldson told them: "If hanging were still an option you would have been executed."
They spent 15 years in prison for the IRA bombing before their convictions were overturned by the Court of Appeal.
It was one of the best known cases of a miscarriage of justice in British legal history and in 2005, Prime Minister Tony Blair apologised to those wrongfully imprisoned.
Mr Conlon's father, Guiseppe, was also jailed as part of a discredited investigation into a supposed bomb-making family, the Maguire Seven, and died - after five years in jail - before his name was cleared.
I believe that we can say, that with all the adversities, in the end Gerry Conlon won - the victory was his.
Gareth Peirce, Gerry Conlon's lawyer
During the service, Mr Conlon's lawyer Gareth Peirce, who campaigned for his freedom, said how he was punished year after year in solitary confinement for protesting his innocence.
"His father, who came to help him, was arrested, convicted and died in prison. His aunt, his uncle, his cousins, their neighbours, were all imprisoned, all on false evidence," she said.
"When he shouted out... what he did and how he articulated it, caused the citadel to fall.
"When he was released, it was euphoric, but it wasn't as [he] believed it would be. Those 15 years had caused damage beyond belief, not just to him, but to the whole of his family and the reunion that they had anticipated every minute of every day for 15 years, wasn't what they thought it would be.
"He fell into an abyss and for years lived like a recluse in the south-west of England where he knew nobody. He couldn't regain the ability to discover joy, he resorted to drugs, to try to find joy."
She added: "Life dealt Gerry a pretty poor hand. He was gambler and gambling was in his DNA but with a poor hand he made a magnificent fist of it.
"If anyone thinks that this was someone who was beaten, terrified, pushed down forever, that wasn't so.
"He fought and he shouted, he was angry, he was awkward, he was uncomfortable, but that was entirely right."
Fr Ciaran Dallat, giving the homily, said Mr Conlon's life had been full of painful memories and carrying a burden of guilt over his father's death.
"He took time and went to other places and tackled other injustices. He reminded us that people get it wrong - institutions, authorities - they get it wrong sometimes," he told the congregation.
Commenting that it may seem unfair that Mr Conlon only had three weeks from his diagnosis of lung cancer until he died, the priest said he "had done an awful lot of suffering in his early life, and he didn't need to suffer again".
As well as his family members, some of those who helped carry his coffin included Paddy Hill, one of the Birmingham Six and Brian Shivers, whose conviction for murdering two soldiers at Massareene Barracks in 2009 was quashed last year.
Politicians who turned out to pay their respects included Sinn Féin representatives, South Belfast MLA Alex Maskey and West Belfast MP Paul Maskey, from the SDLP party leader Alasdair McDonnell, West Belfast MLA Alex Attwood and Foyle MP Mark Durkan as well as Irish Tanaiste Eamon Gilmore.
Mr Conlon was laid to rest at Milltown cemetery after the Requiem Mass.