When the Football League returned following the end of World War II, Dudley made his professional debut in a match against Watford and became a first-team regular at Southend Stadium. He quickly attracted attention from several sides, including Cardiff City who had considered signing Dudley when they completed the signing of his teammate Stan Montgomery in 1948. He eventually joined Leeds United in August 1949 in a player swap deal that saw Albert Wakefield move in the opposite direction. The transfer saw Dudley's weekly wage rise from seven pound per week to twelve pound and, after making his debut for the club against West Ham United on 22 August 1949, he finished his first season with the Yorkshire club as their top goalscorer, scoring 16 times in all competitions, including four goals during their run to the FA Cup quarter-finals. The following season, Dudley continued his goalscoring form, scoring eleven times during the first half of the season before Leeds manager Frank Buckley offered Dudley to Southampton in exchange for Ernie Stevenson, Dudley signing his contract with the "Saints" during a train journey from Leeds to London.