Cordner played for Melbourne University prior to joining the VFL, studying medicine when not on the football field.
A ruckman who was also used in defence, Cordner made his debut for Melbourne in the 1941 finals series, playing in their winning grand final in just his second appearance for the club. Rarely injured, he played 144 consecutive games between 1942 and 1950.
Cordner was Melbourne's best and fairest in 1943 and was a Brownlow Medal winner in 1946, finishing equal sixth the following season. He was club captain in 1948 and 1949, the former in a premiership season. He also captained Victoria in interstate football in the same year. Cordner was also the first Melbourne Cricket Club member to receive the Brownlow Medal.
He retired from football after the 1950 season and went on to become a member of the VFL Tribunal as well as serving on the Melbourne Football Club committee.
In 2000 he was named in the back pocket in Melbourne's official Team of the Century.