The Irishman, who passed away last week aged 90, was assistant manager to Jock Stein when the Hoops became the first British side to win the European Cup after beating Italian giants Inter Milan 2-1 in Lisbon in 1967.
Fallon once spoke to SportTimes about his remarkable career for our Every Picture Tells A Story feature.
And, in a tribute to his life, we reproduce that article in full here.
I joined Celtic in 1950 – and lost myself a lot of money by doing so. I played for the Irish League against the League of Ireland and Mr McGrory (Jimmy, the Celtic manager) approached me after the game.
But they were not the only club after my signature. Fulham, Doncaster Rovers, and Huddersfield were all interested, too. They were offering good wages as well, but it had been a boyhood dream of mine to play for Celtic.
My father was injured in the First World War and had been invalided to Glasgow. During that time he also worked as a postman on the South Side, and he used to go and see Celtic play.
He also knew that Brother Walfrid, who had started the club in 1888, originally came from Sligo in Ireland which was his, and later, my home town. So I had no hesitation in signing from Glenavon for £5,000.
I was earning good money, about £7 a week, working as a confectioner at that time. I was offered just £8, and £6 in the close-season, to go to Parkhead.
I scored the winner against Aberdeen in the Scottish Cup final in 1954, which we won 2-1.
There were 129,000 at that game. It was unbelievable, but if you played for Celtic in those days you were used to it. Once the game started then you forgot about the crowd.
Celtic had been going through a lean period before this win over a good Aberdeen side.
I feel the difference in the turnaround came after we were taken to see the World Cup in Switzerland that same year. It was an education to see a side like Hungary. Their passing, control, and movement off the ball were inspirational.
We changed our system after that and it was so successful we stuck with it. We won the league that season, as well as the cup.
I played at centre-forward for a while. I did so in the League Cup final against Clyde in 1955, which we lost 1-0. But it was a position I always hated.
I think John McPhail was injured and I was switched up front to press the defence alongside Charlie Tully and Bobby Collins. I preferred to play in defence.
The picture shows the Celtic team at the start of the 1955/56 season. We are back row, from left to right: Mike Haughney, myself, John Bonnar, Bobby Evans, Jock Stein and Bertie Peacock. In the front row are, from left to right, Bobby Collins, Willie Fernie, Jimmy Walsh, Charlie Tully and Neil Mochan.
That season I was appointed captain and I chose Jock as my deputy. I broke an arm three times that term, though, and Jock had soon taken over the role from me. He was a good player and could read situations brilliantly. He was very strong in the air as well. He was a great trainer.
But that was to be his last season as a player. He injured an ankle playing against Rangers in a challenge with their striker, Don Kichenbrand, and had to call it a day.
Supporters may not realise this, but players on opposite sides of the Old Firm want to beat each other on the park, yet away from it are great pals.
Big Corky (legendary Rangers captain George Young), for example, couldn't do enough to help Celtic players.
He got jobs for a lot of Celts in Glasgow after they had finished their playing careers.
I remember one match against Rangers well. I elbowed him in the nose and it started to bleed. He stood up and shouted: ''I'll kill you, I'll kill you.'' But it was all forgotten about afterwards.
I only had one manager, Mr McGrory, during my time with Celtic as a player and I thought he was a very nice man. Maybe a bit too nice. He hated to hurt people. Sometimes you have got to be hard on players who take advantage of you.
At that time managers never went on to the training field with their players. That was a shame, as Mr McGrory had so much to offer in that department. He was a great player with Celtic. Some of the goals he scored were just unbelievable.
I can remember just one time when he came on to the pitch at a training session. He was in his suit and the ball was crossed in his direction. He put his head on to it and it went into the goal like a bullet. He said: ''That's how you do it.''
It was a pity he didn't do it more often. I think it was the chairman, Bob Kelly, who picked the teams back then. He ran the whole show.
I finished off my playing days in 1958 after suffering a bad knee injury. I was experiencing excruciating pain and had to have cortisone injections. I was still very fit, but my surgeon advised me to retire.
The club had just bought their Barrowfield training ground and asked me to take over the running of the third team and take care of bringing players to the club.
I brought back Willie Fernie to help me. Jock was in charge of the reserve team.
The chairman was friendly with Jock back then and he thought it would be good for him to go away and get experience at another club. It is difficult to go from being a player straight to coaching your team-mates.
So Jock went away to Dunfermline, and then moved on to Hibs, and I moved up to take charge of the reserve team.
After a few seasons, though, Jock was approached to be Celtic manager and he specified that he wanted me as his assistant.
I took a few players to the club who enjoyed great success. We were having problems with goalkeeping, so I snapped up Ronnie Simpson. He already had two FA Cup winners' medals with Newcastle United.
I also thought former player Bertie Auld, who was at Birmingham at that time, would be able to help us. We got him for £10,000 and he made a huge difference. He made Bobby Lennox a better player.
But we had a great team back then, with Tommy Gemmell, Jim Craig, Billy McNeill, John Clark, as well as Jimmy Johnstone and Stevie Chalmers, to name but a few.
Jock was never happy, though. He was always looking for 100 per cent. He was very strong on discipline as well. He played such an important part in our 2-1 triumph over Inter Milan in the European Cup final in 1967.
When we returned, Parkhead was quite a scene with the ground packed and everybody jubilant. I was so pleased for the supporters that we had managed such an achievement, especially after the tough times they had had for so long.
We won every tournament we entered that year – the League, League Cup and Scottish Cup. That will never be repeated.
Jock had been involved in a car crash at the beginning of the 1975 season, so I took charge for that campaign.
We had some great young players coming through at that time, with Kenny Dalglish, Vic Davidson, Roy Aitken, Lou Macari, Danny McGrain and Pat Bonner all involved.
I had been responsible for signing a lot of them. Some of those guys went on to become great players at club and at international level. But at the time it was impossible to tell who would make it.
Sometimes they had the ability, but didn't look after themselves properly.
In 1978, Jock was replaced as manager by Billy McNeill. I think he felt he had achieved all that he could. I also left the club and joined Davie Wilson at Dumbarton and spent two years there.
You learn all about what hard work is at a club like that. People who run the smaller clubs should get a great deal of credit. Davie was marvellous as well, full of enthusiasm.
I had been friendly with the Dunn family, who ran Clyde since I first came to Scotland, and they asked me if I would like to go on the board of directors. So I did. But the travelling to Cumbernauld, as well as away matches to places like Stranraer and Inverness, was getting to me, so I stood down.
I kept in touch with a lot of the players I was involved with at Celtic. I was honoured to be invited back to Parkhead to unveil the League Championship flag at the start of the 2012/13 season.