Archbishop Tutu, who helped end apartheid in South Africa, died in Cape Town on Boxing Day.
The First Minister said: “Such sad news this morning…but his was a life that made the world a better place.
“Rest in peace, Archbishop Desmond Tutu.”
Humza Yousaf, Scotland’s Health secretary also paid tribute, commenting on Tutu’s fundamental role in the anti-apartheid movement.
He said: “Such sad news to wake up to this morning, but also the opportunity to celebrate the life of an incredible man.
"A giant of the anti-Apartheid movement. A global moral compass on so many issues, much needed in the times we live in.
"Rest in peace and power.”
Piyushi Kotecha, chief executive of the Desmond and Leah Tutu Legacy Foundation, and chairman Niclas Kjellstrom-Matseke said in a statement that Tutu was “a living embodiment of faith in action”.
In a statement on the foundation’s website, they added he spoke “boldly against racism, injustice, corruption and oppression, not just in apartheid South Africa but wherever in the world he saw wrongdoing, especially when it impacted the most vulnerable and voiceless in society.”
According to the trust, he died peacefully at the Oasis Frail Care Centre in Cape Town. A cause of death has not been given.
Nicknamed “The Arch”, Tutu was made the first black Anglican Archbishop of Cape Town in 1986 and was a driving force to end the policy of racial segregation and discrimination in South Africa from 1948 until the early 1990s.
Tutu is known for the quote, “If you are neutral in situations of injustice, you have chosen the side of the oppressor.”
His work led to him receiving numerous doctorates and academic awards from all over the world. He retired from public life in 2010 yet continued to do charity work through the Desmond and Leah Tutu Legacy Foundation.
South African President Cyril Ramaphosa said it was “another chapter of bereavement in our nation’s farewell to a generation of outstanding South Africans who have bequeathed us a liberated South Africa.”
“Desmond Tutu was a patriot without equal; a leader of principle and pragmatism who gave meaning to the biblical insight that faith without works is dead,” he tweeted.
“We pray that Archbishop Tutu’s soul will rest in peace but that his spirit will stand sentry over the future of our nation.”
The Nelson Mandela Foundation, which highlighted the friendship between the pair, said the loss of Tutu is “immeasurable”.
Justin Welby, the Archbishop of Canterbury, described Tutu as “a man of words and action”.
Prime Minister Boris Johnson said Tutu would be remembered for his leadership and humour.
He said: “I am deeply saddened to hear of the death of Archbishop Desmond Tutu.
“He was a critical figure in the fight against apartheid and in the struggle to create a new South Africa – and will be remembered for his spiritual leadership and irrepressible good humour.”