The Irishman passed away peacefully in his sleep in the early hours of Friday morning.
Fallon won the Scottish Cup twice in 1950 against Motherwell and two years later against Aberdeen, when he scored in the Hampden final.
He also starred in the 7-1 League Cup final rout against Rangers in 1957 when his replacement as skipper Jock Stein lifted the trophy.
And the Sligo-born man was Stein's assistant as Celtic swept all before them during the club's most successful period from 1965 to 1977.
Aside from the on field triumphs which saw the club crowned Champions of Europe in 1967, Fallon was instrumental in signing the likes of Kenny Dalglish, Danny McGrain, David Hay and Pat Bonner.
Fallon unveiled the Premier League flag on the opening day of the current season, when Celtic beat Aberdeen 1-0.