Among those paying their respects were players Fallon, known for his keen eye for talent, signed for the Glasgow club, including Lisbon Lions Tommy Gemmell and Bertie Auld, Kenny Dalglish, Pat Bonner, David Hay and Danny McGrain.
Manchester United boss Alex Ferguson paid a fond tribute to his long-standing friend, saying he had learned from him as a player and manager.
The Requiem Mass was held in Fallon's parish at Christ the King Church in King's Park with a mass to mark his passing also said at 10am in his home town of Sligo, Ireland.
The former player and assistant manager died peacefully at the age of 90 last Friday.
Celtic boss Neil Lennon attended the service with many of his players, including Scott Brown and Georgios Samaras, with Rangers manager Ally McCoist also in attendance.
Fallon's wife of 54 years, Myra, was accompanied by daughters Marie Therese, Louise Ann, Collette, Siobhan and Sinead and son Sean.
He had 19 grandchildren, with another on the way. His sisters, Marie and June, were also present.
Sir Alex told mourners of how he first met Fallon while playing for Rangers. He said: "I first came into contact with him in the Beechwood. Jock (Stein) and Sean would be there, I would pump him for knowledge. I was at Rangers at the time.
"I told them I'd love to come and watch a training session. They both looked at one another and that was their answer. When Jock became manager ... I use an assistant manager in the same way.
"It's about getting to the heart of the dressing room, knowing what's going on. What he did for Celtic in bringing through those players was fantastic."
Other Lisbon Lions who came to pay their respects included Stevie Chalmers, Jim Craig, Bobby Lennox, John Clark and John Hughes.
Honouring his allegiance to Celtic, Father Desmond Keegan said: "Like the huddle, so many people look to put their arms around the Fallon family.
"With a huge congregation like this it reminds Myra that you will never walk alone."
The priest recounted how he first met Fallon when he was at primary school with his daughters.
The footballer organised a trip to Parkhead for the school and spoke with "pride and passion" about the club and its roots with the poor.
The priest invited his grandchildren to lead the bidding prayers on the altar, in front of the coffin which was simply draped in white.
Son Sean spoke of a "humble and generous" father who never turned down a charity request and did all he could to nurture young footballers.
He told how Lou Macari gave him his boots from the 1978 World Cup because "your father was so kind to me".
He said the legacy of Celtic's European Cup win would remain with the Fallon family for generations.
A private burial followed at Philipshill Cemetery in East Kilbride.