He scored the winning goal for St Mirren on his debut, a 2–1 victory against a Hibernian team containing his brother Joe. On 25 April 1959, he scored the third goal in St Mirren's 3–1 victory against Aberdeen in the 1959 Scottish Cup Final. That was his eighth goal of the 1958–59 Scottish Cup, as he scored in every round. On 30 January 1960, he scored ten goals in a 15–0 win over Glasgow University. Baker was the club's top goalscorer in both the 1958–59 and 1959–60 seasons.