Mr Blair, who was with his father when he passed away earlier today, said he was "privileged to have him as a Dad".
The former PM - who pulled out of an engagement with former US president Bill Clinton last night after his father fell ill - said: "He was a remarkable man. Raised in a poor part of Glasgow, he worked his way up from nothing, with great ambitions dashed by serious illness on the very brink of their fulfilment.
"He lost my mother, whom he adored, when she was still young. Yet despite it all he remained animated by an extraordinary spirit that was in him until the end.
"I was privileged to have him as a Dad."
Leo Blair, the illegitimate son of two English actors, was adopted as a baby by Glasgow shipyard worker James Blair and his wife, Mary.
His second son, Tony, now 59, was born in Edinburgh in 1953 and the future Prime Minister spent the first 19 months of his life at the family home in the Willowbrae area of the city while his father worked as a junior tax inspector and studied for a law degree from Edinburgh University.