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."
Lawwell continued: "He did so with fearless distinction over eight years in the 1950s, and Sean was at the heart of all of our triumphs, including that famous 7-1 League Cup victory in 1957.
"Ten years later, he was part of our greatest ever success, winning the European Cup in Lisbon.
"Working alongside Jock Stein, Sean helped to deliver that trophy, along with nine consecutive league titles and numerous other trophies.
"He was an integral part of the club's success, and also played a vital role in identifying talented young players who would go on to become great Celts.
"He was a gentleman with a great sense of humour, and always a welcome guest here at Celtic Park. And there could have been no more fitting person to unveil the championship flag at this start of this, our 125th anniversary season, than a man who gave so much to Celtic Football Club.
"Sean will be sadly missed but he will never be forgotten for what he did for our club."
Manager Neil Lennon added: "We are all shocked and saddened to hear of Sean's death, and the thoughts of all the players and management are with his family.
"Sean was a great Celtic man, and it's been a privilege to meet him over the years. He was always a very humble man when you met him, even though his achievements in the game were extraordinary.
"He did everything you could ever wish to achieve in football, as a player for both club and country, while he went to even greater things in management alongside Jock Stein.
"For these things, the Celtic supporters have always loved Sean and recognised him as one of our own because it was always his great love of Celtic which shone through.
"It was a fitting tribute that Sean unveiled the championship flag back in August and the welcome he received from the supporters spoke volumes. We will all miss him."