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 to secure a league and cup double for the Hoops.
A member of the famous '7-1 team' who beat Rangers in the 1957 League Cup Final, Fallon scored 14 goals in 254 games for Celtic while representing the Republic of Ireland eight times.
The Sligo-born defender was Jock Stein's assistant during the club's most successful period from 1965 to 1975 and was also responsible for leading skipper Billy McNeill through a throng of fans to collect the European Cup trophy in 1967.
As Stein's righthand man, Fallon was instrumental in bringing some top names to Celtic Park including Danny McGrain, Davie Hay and Pat Bonner, plus legend Kenny Dalglish whom he persuaded to join the club instead of boyhood heroes Rangers.
Fallon, a regular visitor to Parkhead, unveiled the Premier League flag on the opening day of the current season, when Celtic beat Aberdeen 1-0.
Chief executive Peter Lawwell told the club's website: "Sean Fallon was a Celtic legend who devoted much of his life to the club he loved.
"He was a supporter, first and foremost, and he was proud to have worn the green and white hoops that he loved."