Mr Smith, who was born in Barnsley, was a critic of austerity and championed human rights and the welfare state - his speech at the Labour Party conference in 2014 when he talked about life before the NHS moved many listeners to tears.
A post on his Twitter account, which has more than 250,000 followers, announced his death in a hospital in Canada on Wednesday morning, following an illness.
The tweet said: “At 3:39 this morning, my dad Harry Leslie Smith died. I am an orphan. #istandwithharry”.
Labour leader Mr Corbyn shared a video clip of Mr Smith speaking about the importance of the NHS, and wrote: “We will all miss Harry Leslie Smith - he was one of the giants whose shoulders we stand on.
“A World War Two veteran who dedicated his life to fighting for our National Health Service, a peaceful world and for countries to meet their moral responsibility by welcoming refugees.”
Deputy Labour leader Tom Watson paid tribute, writing on social media: “Harry Leslie Smith will remain an inspiration to all in the Labour movement engaged in the fight for justice and fairness.
“The world is a far better place for his life, words and deeds; and a far sadder place with his loss. Farewell @Harryslaststand”
Former Labour leader Ed Miliband also tweeted: “Very sad to hear of the death of Harry Leslie Smith. He was one of a kind who never wavered in his fight for equality and justice. We should all carry his passion, optimism and spirit forward.”
Mr Smith described himself as “the world’s oldest rebel” and dedicated his later years to highlighting social injustice and poverty around the world.
Mr Smith, an RAF veteran, survived the Great Depression and Second World War, and rose to prominence in 2013 after writing an article in the national press declaring that he would no longer wear a poppy.
The son of a coal miner, Mr Smith has spoken of growing up in the “barbarous” and “bleak” time of Barnsley during the 1920s, saying “it was an uncivilised time because public healthcare didn’t exist”, before warning that “we must never ever let the NHS free from our grasp because if we do your future will be my past”.
He suffered the loss of his sister Marion at the age of three from tuberculosis and would go on to support his family aged seven when he took up work as a barrow boy for a brewery in Bradford.
He called voting for Labour and the creation of the health service after the 1945 general election one of the proudest moments of his life.
The global financial crisis of 2008 prompted Mr Smith to take a “last stand” against what he saw as the excesses of capitalism and erosion of public services.
His subsequent book, Harry’s Last Stand, was released to widespread acclaim in 2014.
Calderdale plaque proposal
Councillors in Calderdale are among those that have paid tribute to Harry Leslie Smith and are exploring the possibility of putting up a blue heritage plaque in Boothtown, Halifax, where he lived for a time during his youth.
Sowerby Bridge ward councillor Adam Wilkinson said Mr Smith had been an inspirational figure who had moved to Halifax from Bradford in 1937 – Harry’s family moving from his Barnsley birthplace in South Yorkshire to West Yorkshire when his miner father became unemployed.
Council leader, coun Tim Swift said he would be looking into the possibility of putting up a blue plaque to Mr Smith.