Fallon made more than 250 appearances for Celtic during the 1950s and was assistant manager to Jock Stein throughout the most successful spell in the club's history.
The club announced on their Twitter feed: "Celtic legend Sean Fallon has died peacefully at the age of 90. Our thoughts and prayers are with his family."
The Irishman unfurled the Scottish Premier League flag before the champions began the defence of their title at Celtic Park in August last year.
Born in Sligo on July 31 1922, Fallon played for Celtic between 1950 and 1958, making 254 appearances for the club and scoring 14 goals.
He made his league debut for Celtic in the last game of the 1949/50 season and won his first piece of silverware by helping the club beat Motherwell in the Scottish Cup final in 1950/51. He became captain in October 1953 but was forced to retire through injury in 1958. However, he remained an influential part of the club and became Jock Stein’s assistant in 1965.
As assistant manager to Jock Stein, he was instrumental in helping Celtic secure nine league championships in a row and the European Cup on May 25, 1967.
Fallon also brought names including Tommy Gemmell, Davie Hay, Danny McGrain, Kenny Dalglish, Lou Macari and Packy Bonner to the club.
He left Celtic in 1978 and became assistant manager of Dumbarton before becoming manager in 1980.
Celtic chief executive Peter Lawwell said: “This is very, very sad news and our thoughts are with Sean’s wife, Myra, their children Marie-Therese, Louise, Collette, Siobhan, Sean and Sinead, and all of Sean’s family and many friends.
“Sean Fallon was a Celtic legend who devoted much of his life to the club he loved. He was, of course, a supporter, first and foremost, and he was proud to have worn the green and white Hoops that he loved.
“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.”
Celtic manager, Neil Lennon said: “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.”
Sports broadcaster Dougie Donnelly paid tribute to Fallon on Twitter.