The widely respected former neurosurgeon, who went on to become a Labour MP and MSP for Strathkelvin and Bearsden in Glasgow, had the transplant in 1990 at the Freeman Hospital in Newcastle.
He was education minister at the time of the exams crisis in 2000 when thousands of students received incomplete or inaccurate results certificates from the Scottish Qualifications Authority. He later became environment minister before quitting politics in 2001 for health reasons.
Mr Brown said Mr Galbraith, a Scottish Office minister in Tony Blair's first Labour Government, was great in every respect: "A great surgeon, a great statesman and a great family man who will be sorely missed.
"For him politics was about one thing only - service to the people. Our thoughts are with his wife, Nicola, and three daughters, Mhairi, Heather and Fiona."
Former Chancellor Alistair Darling added that the ex-MSP, who died yesterday morning, had been a "brilliant neurosurgeon, a dedicated politician" and a dear friend who was devoted to his family.
Mr Darling said it was "extraordinary" how Mr Galbraith had contributed so much in the 25 years he had lived after the lung transplant.
He added: "Sam's professional life and immense talents were devoted to the care and betterment of others, through the National Health Service and in his deep political commitment. He was a great humanitarian. His work and his life touched countless lives."
First Minister Alex Salmond said: "Sam Galbraith was a devoted and gifted politician whose commitment to improving the lives of others never wavered, even though he faced living with his own serious health condition for the best part of 25 years. My thoughts go to his family and friends at this sad and difficult time."
Holyrood Presiding Officer Tricia Marwick said there was always a sense that the parliament was "robbed of his talents" when he stood down in 2001.
Scottish Labour leader Johann Lamont said: "Sam was a great champion of the NHS, and he himself was a great example of how it can transform lives."
Labour's shadow foreign secretary Douglas Alexander said he would miss him, adding he was "a good man".