Sir Rex will be remembered for his “courage and dignity” in facing the invasion, the Falkland Islands government said.
He also served for many years as chairman of the Falkland Islands Association and as president of the UK Falkland Islands Trust and was granted the freedom of the island capital, Stanley, in 1985.
The Yorkshireman took up his post in 1980 and was ousted as governor of the UK overseas territory by invading Argentinian forces on April 2 1982.
After refusing a plan to hide in the hills and outnumbered after a fierce gun battle, Sir Rex took the painful decision to surrender.
But he made one final act of defiance when, dressed in full gubernatorial uniform, he refused to shake invading commander General Oswald Garcia’s hand.
The islands were recaptured by UK forces on June 14 that year after then-Prime Minister Margaret Thatcher deployed a naval taskforce to the South Atlantic ocean.
During the weeks of the war, Sir Rex stayed in London while his wife and son were in a house in Kent.
Yesterday the Falkland Islands government said in a statement: “We received news of the passing of Sir Rex Hunt with great sadness.
“Sir Rex will forever be remembered in the islands for his years of service as governor, and particularly for his courage and dignity in facing the Argentine invasion in 1982.
“His numerous contributions to our home were recognised in his being granted the freedom of Stanley in 1985.
“A loyal friend of the Falkland Islands, he served for many years as chairman of the Falkland Islands Association and as president of the UK Falkland Islands Trust.
“His passion and commitment to the Falkland Islands will be sorely missed. The thoughts and deepest sympathies of all Falkland islanders are with his family and friends at this sad time.”
In a 2007 interview, Sir Rex said his first thoughts on the liberation day were for the soldiers.
He said: “I was thinking of what had happened at Government House in April.
“To see all those people, hundreds of them, throwing their guns away with no attempt to fight, it just made me think ‘Well thank goodness, we’ve got people like our army’.”
Foreign Secretary and Richmond MP William Hague paid tribute to Sir Rex, saying he showed “great courage and fortitude” during the Argentine invasion.
Mr Hague said: “I was very sorry to hear of the death of Sir Rex Hunt. As governor of the Falkland Islands he served his country and the Falkland Islanders with great distinction. Like the Islanders themselves, he demonstrated great courage and fortitude in the face of Argentine aggression during the Falklands conflict.
“After the conflict, he contributed to the transformation of the Islands into a vibrant, modern and successful democracy.
“The warm tribute the Falkland Islands government has paid him today reflects the genuine respect, gratitude and affection felt for Sir Rex by the people he worked so hard to support. His contribution will not be forgotten.