Mr Galbraith, 68, who is believed to have been the world's longest-surviving lung transplant recipient, died in hospital in Glasgow from an infection.
The former neurosurgeon, who went on to become a Labour MP and MSP for Strathkelvin and Bearsden, received the transplant in 1990 at the Freeman Hospital in Newcastle.
The father of three 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.
Later that year he was moved in a cabinet reshuffle, becoming environment minister.
Mr Galbraith resigned from politics in 2001 for health reasons.
Former chancellor Alistair Darling said he had been asked to put out a statement on behalf of Mr Galbraith's family, which said: "Sam Galbraith died this morning in Glasgow's Western Infirmary after contracting an infection which proved impossible for him to overcome.
"Sam was a brilliant neurosurgeon, a dedicated politician and a very dear friend. But above all, he was devoted to his family, to Nicola and their three daughters - Mhairi, Heather and Fiona - who are foremost in our thoughts.
"They have asked for their privacy to be respected at this time."
It continued: "Sam was believed to be the world's longest-surviving lung transplant patient.
"It was extraordinary how he continued to contribute and achieve so much throughout the 25 years that he lived with his condition.
"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.
"Sam's family wish to thank all the staff in the medical assessment unit and intensive therapy unit of Glasgow Western Infirmary for their care and attention in his last illness, and also his GP and all the staff in the transplant unit at Freeman Hospital, Newcastle, for their magnificent care over many years."
Scottish Labour leader Johann Lamont also paid tribute to him and said: "Sam was a wonderful man. A great doctor, an inspiring colleague and dedicated family man.
"Sam was a great champion of the NHS, and he himself was a great example of how it can transform lives.
"We will all miss his vigour, his support and his candour. Our thoughts are with his wife and family."