Born in Fife, Connelly was a technically accomplished footballer, who could play with distinction anywhere in the outfield, although his most common positions were left half and inside forward. He signed for Celtic from Tulliallan Juniors in June 1965 and as a teenager was noted for his fine ball control. This was first publicly displayed when he was sent out to entertain the crowd at Parkhead before a European tie later that year against Dynamo Kiev. He was considered by many to have the potential to be a world class player – as influential in British football as Beckenbauer was in the German game. Having broken into the first team in 1968, Connelly is remembered primarily for two goals. Just before half time in the 1969 Scottish Cup Final against Rangers, he coolly dispossessed John Greig on the edge of the box, evaded the Rangers’ skipper’s recovery attempt, rounded the goalkeeper before slipping the ball into the empty net. This goal made it 3–0 to Celtic and ended any hopes of a Rangers revival.
In 1970, in a game that was referred to by the contemporary media as the football “Battle of Britain”, he scored for Celtic in a 1–0 win against the English champions Leeds United. His first-minute strike in the first leg of the Champions’ Cup semi final at Elland Road helped Celtic progress to their second European Cup final, against Feyenoord.