Gallagher made his debut for Celtic on 22 August 1958 during a 1–0 victory over Raith Rovers F.C. in the Scottish League Cup. He then made his full debut on 6 March the following year, when Jock Stein signed him as to a professional contract that same day. Gallagher took the corner kick that resulted in Billy McNeill's winning goal in the final of the 1965 Scottish Cup. Two years later, he was made captain for a Scottish Cup game against Elgin City. This was in tribute to his accomplishment of playing for the Republic of Ireland, becoming the first player born in Scotland to represent the team when he played against Turkey on 22 February 1967. Gallagher was part of the Lisbon Lions squad that won the European Cup Final that same year, which saw him score four goals in the run-up to the final. Although he did not start in the final, the club maintained that he "has always been rightly acknowledged as part of the Lisbon Lions squad".