Born in Carluke, Lanarkshire, Dodds joined Celtic in the summer of 1908 from Carluke Milton Rovers. He made 351 league appearances. Dodds was considered among the Celtic greats of his era and along with Shaw and Alec McNair was considered to have formed one of the greatest club defences in British football. At Celtic Dodds picked up many senior medals.
While at Celtic, Dodds gained three full Scotland caps, including playing in the 3–1 win over England on 4 April 1914. This followed draws against Wales (0–0 at home on 28 February) and Ireland (1–1 away on 14 March) of the same year, giving Dodds an unbeaten full international record. Dodds also represented the Scottish League XI eight times between 1912 and 1920.