Cordon was born in Cuero, Texas. He moved with his family to Roseburg, Oregon as a child, and attended the public schools there. In 1909, at the age of 19, he became the deputy tax assessor of Douglas County, Oregon. He remained in that position until 1916 when he became the county tax assessor. He was the tax assessor until 1920 when he decided to become a lawyer.

Cordon served as the district attorney of Douglas Country, Oregon from 1923 to 1935. He then practiced law in Roseburg.

In 1944, Cordon was appointed to a seat in the United States Senate from Oregon by Governor Earl Snell following the death of Senator Charles L. McNary. In a special election in November 1944, Cordon was elected to finish the term, receiving 57f the vote against Democrat Willis Mahoney. Cordon was elected to a full six-year term in 1948, receiving 60f the vote against Democrat Manley J. Wilson. In 1954, a bad year for Republicans, Cordon was defeated for re-election by Democrat Richard L. Neuberger by a margin of 50.2 o 49.8åP> While in the Senate, Cordon suggested a rule (now known as the Cordon Rule) that Senate committee reports should indicate how provisions in a bill would change current law. He visited Hawaii to conduct hearings on possible statehood for the then-territory (Hawaii was admitted in 1959). Cordon served as chairman of the Committee on Interior and Insular Affairs from 1954 until his term expired in 1955.

Cordon practiced law in Washington, D.C. from 1955 to 1962, when he retired. He died in Washington, D.C. and was buried in Roseburg.