Mr Eadie, who was the constituency’s MP for 26 years until his retirement in 1992, served as a junior minister in the governments of prime ministers Harold Wilson and Jim Callaghan.
He died at his home in East Wemyss, Fife, after a short illness and is survived by his wife Janice and son Bob Eadie, a Labour councillor in Fife. His daughter-in-law, Helen Eadie, is Labour MSP for Cowdenbeath.
Former prime minister Gordon Brown said: “This is very sad news. Alex served diligently both as a Fife councillor then as a much-loved MP for Midlothian for a quarter of a century.
“He distinguished himself as a minister in Harold Wilson and Jim Callaghan’s governments. Our thoughts are with all of his family.”
Educated at Buckhaven Senior Secondary School in Fife, Mr Eadie worked as an agent for the National Union of Miners and served on the former Fife County Council for 20 years before he was elected MP for Midlothian in 1966, having stood in Ayr at the 1959 and 1964 general elections.
Appointed in 1972 by the then-Labour leader, Harold Wilson, as opposition spokesman on energy, he became minister for coal after Labour won the February 1974 election and held the post for five years.
He was disappointed not to be called at the last Scottish question time in the Commons before his retirement at the 1992 general election. When he complained, the then-speaker Bernard Weatherill invited him for a dram as a consolation.
Current Midlothian MP David Hamilton said he had first met Mr Eadie as a pit delegate at Monktonhall colliery in the mid-1970s.
He said: “Alex was minister for coal and he came to Monktonhall to launch the major investment when they were talking about putting tunnels out under the Forth. He was a really good character.
“When I made my maiden speech, I referred to Alex and he sent me a nice note afterwards saying ‘It was a lovely speech, but I was there a year longer than you said’.”
Scottish Labour leader Johann Lamont said: “Alex Eadie was a great socialist, co-operator and Member of Parliament for Midlothian. Starting out as a miner, he was a mainstay of the National Union of Mineworkers before being elected to local government in Fife and then going on to be the MP for Midlothian.
“He served in both the Wilson and Callaghan governments but his first priority was always those of his constituents.
“He will be remembered very fondly by the Labour colleagues, who always respected his sage advice and down-to-earth manner.
“Even in his later years he remained an enthusiastic and passionate political debater and fan of his football team, East Fife. Our thoughts are with Alex’s family and friends.”