Fallon signed for Celtic from junior football team Fauldhouse United in 1959 to understudy for Frank Haffey and made his debut that same year against Clyde. He replaced Haffey as first choice goalkeeper in 1963 and played his part in the UEFA Cup Winners Cup run of 1964, where they lost out to the Hungarian team MTK Budapest 4–3 in the semi-finals, after holding a 3–0 lead from the first leg at Celtic Park. He remained first choice goalkeeper until the signing of veteran keeper Ronnie Simpson from Hibernian in 1965.
Fallon became the only Lisbon Lion not to play in Celtic’s win in the 1967 European Cup Final against Inter Milan, as he was the (unused) substitute goalkeeper. At that time the only substitute permitted was for the goalkeeping position. As a result of winning the European Cup, Celtic entered the Intercontinental Cup later that year, a two-leg match against Racing Club of Argentina. While warming up for the second leg, a brick (Sir Robert Kelly says it was a “flat iron bar”) thrown or catapulted by a member of the crowd, struck Ronnie Simpson on the head.
With Simpson unable to play, Fallon was called on to keep goal in a bad-tempered second leg, and an even more bad-tempered play-off. Although Celtic lost the return leg, and the subsequent play-off in Uruguay, amid accusations of extreme foul play from the Racing players, Fallon distinguished himself with a string of saves which caused one commentator to remark: “If that’s the reserve keeper, what must the other guy be like?” According to Sir Robert Kelly in his book Celtic, when Celtic tried to get the second leg declared void because of the Simpson incident, Racing Club used Fallon’s heroic performance as their counter argument, and the result stood. Fallon also played in the team that won the Alfredo di Stefano Trophy, a testimonial for the Real Madrid player.
After Simpson’s retirement, Fallon took over as first choice at Celtic. He lost his place to Evan Williams due to a long illness, sitting out the 1970 European Cup Final. He later played for Motherwell and Morton before retiring from senior football.