Connelly began his playing career with
St Helens Town in the
Lancashire Combination. In November 1956,
scouts from
Burnley came to watch another player, but were so impressed by Connelly that he was offered a trial with Burnley, and subsequently made a permanent move to the club, who were then playing in the English top division. He made his debut away from home against
Leeds United on 11 March 1957. It took two seasons for Connelly to gain a permanent place in the Burnley first team, which he did in the
1958–59 season, scoring 12 goals from 37 appearances, and ending the season as the team's second top scorer. He was predominantly a right winger who could also play on the left. He had the ball control and speed to beat the
full back on the outside and deliver accurate crosses, but was also able to cut inside and score goals with either foot. For a winger, he was always a prolific goal scorer, scoring 105 goals for Burnley in 265 appearances. In the
1959–60 season, Connelly played a crucial part in helping Burnley to end as league champions for only the second time. He scored 20 goals in 34 league appearances. Unfortunately for him, he missed the vital last game against
Manchester City that gave Burnley the title. His replacement,
Trevor Meredith, scored the final winning goal while Connelly was in hospital having a cartilage operation. However, he still received a Championship winner's medal. In the following three seasons, Connelly played a full part and scored plenty of goals to help Burnley finish fourth, second and third, respectively, in the league. In addition to finishing second in the
1961–62 season, Burnley reached the
FA Cup final where they were beaten 3–1 by
Tottenham Hotspur. Connelly played on the right wing and received a runners-up medal. In the
1963–64 season, the emergence of a talented young Scottish winger,
Willie Morgan, meant that Connelly was forced to play on the left wing. In April 1964 he was transferred to
Manchester United for £56,000. ==Manchester United==