The former miner, who served in the Wilson and Callaghan governments, died at his home in East Wemyss, Fife, after a short illness.
He is survived by his wife Janice and son Bob, a Labour councillor in Fife. His daughter-in-law is Helen Eadie, 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 almost 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.”
Scottish Labour leader Johann Lamont said: “Alex Eadie was a great socialist, co-operator and member of parliament for Midlothian. 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 … governments, but his first priority was always 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 at this time.”