The Galway man served as GAA president between 1997 and 2000 after winning an All-Ireland senior hurling medal and an All-Star during his playing days.
McDonagh helped Galway win the National League title in 1975 and earned his All-Star a year later.
After captaining Galway during their All-Ireland final defeat in 1979, he won a Liam McCarthy medal a year later.
McDonagh didn't start in the 1980 All-Ireland Final win over Limerick but he memorably led the rendition of the West's Awake on the steps of the Hogan Stand following Galway's emotional triumph.
His GAA presidency saw him leading the calls for the removal of the controversial Rule 21, which barred members of the British security forces from playing gaelic games.
The rule was eventually abolished a year after his presidential term concluded.
Current GAA president Aogan O Fearghail led the association in paying tribute to the Galway man.
"It was with great sadness that we learnt of Joe's passing last night and I know that sentiment is shared by so many throughout the wider GAA family," said the GAA president.
"Joe was held in extremely high regard and his company was enjoyed by so many over the course of his long involvement with the GAA in so many different capacities, not least as president from 1997 to 2000.
"He and his tenure left an indelible mark on Cumann Lúthchleas Gael and his passion for our games and activities was only matched by his passion for the Irish language."
McDonagh's passing comes two weeks after the death of his predecessor as GAA president Jack Boothman.
Former GAA president and ex-Galway hurling star Joe McDonagh has died after a short illness at the age of 63.