Mr Conlon spent 15 years in prison for the 1974 attack which killed five people and injured 65, before the convictions were overturned by the Court of Appeal in 1989.
In 2005 Prime Minister Tony Blair apologised for the miscarriage of justice.
The Belfast man had been ill for some time before passing away in his Falls Road home.
He was 60.
Mr Conlon's family issued a statement through his lawyer Gareth Peirce.
It said: "This morning we lost our Gerry.
"He helped us to survive what we were not meant to survive.
"We recognise that what he achieved by fighting for justice for us had a far, far greater importance - it forced the world's closed eyes to be opened to injustice.
"It forced unimaginable wickedness to be acknowledged.
"We believe it changed the course of history. We thank him for his life and we thank all his many friends for their love."
He brought life, love, intelligence, wit and strength to our family through its darkest hours.
Writing in the Guardian in 2009 about the introduction of anti-terrorism laws, Mr Conlon said he suffered nightmares about his time in prison following his release.
He said how he was beaten constantly and suffered abuse at the hands of both prison staff and fellow inmates.
"It was almost as if I was in the eye of the storm while I was inside, and everything was being held back for a replay later in my life," he wrote.
He detailed how he suffered two breakdowns, attempted suicide and became addicted to drink and drugs following his release.
"The ordeal has never left me. It was a terrible price to pay," he said.
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.
He was arrested while travelling to London from Belfast to help his son.
His mother Sarah, a tireless campaigner for their freedom, died in 2008, aged 82.
The Oscar nominated film, In The Name Of The Father, was based on the family's wrongful incarceration and bid for freedom.
Daniel Day-Lewis played the role of Gerry Conlon.
SDLP MLA Alex Attwood paid tribute to Mr Conlon who had been a supporter of his party.
He said: "He'd given an awful lot but yet had so much more to give.
"He's now with his dad and his mum."
What he learned from his time in prison and campaign for release was the importance of not only raging against his own injustice but fighting for those who had also suffered miscarriages of justice.
Alex Attwood MLA
Sinn Féin president Gerry Adams expressed his shock and deep sadness at the news.
"Gerry and his father Giuseppe were two of the most infamous examples of miscarriages of justice by the British political and judicial system," he said.
"To his family and friends I want to extend my sincere condolences."
Ireland's Foreign Affairs Minister, Eamon Gilmore, also extended his condolences.
"I am saddened to hear of the death of Gerry Conlon and send my condolences to his family and friends," the Tánaiste said.
"Mr Conlon suffered a grave miscarriage of justice along with his father, Giuseppe, Paul Hill, Carol Richardson and Paddy Armstrong.
"In later years Gerry drew from his experiences to campaign on behalf of others with the group Miscarriages of Justice Organisation.
"His loss will be felt both within the community in west Belfast and across the world with all those who work in pursuit of justice."