Cornelius was born in East Newark, New Jersey on December 10, 1878. Throughout his younger years he attended Holy Cross School in Harrison, New Jersey, and St. Francis Xavier’s High School in New York City. C. A. McGlennon later attended college and graduated from Seton Hall College in South Orange, New Jersey in 1899. After this, he became a public and high-school principal from 1901 until 1926. Cornelius also studied law at the New Jersey Law School in Newark, New Jersey before he was admitted to the bar in 1916. McGlennon practiced law in East Newark, New Jersey. Sometime afterwards, McGlennon's interest peaked in politics. McGlennon served as member of the State senate in 1917 and 1918, and then served as Democratic floor leader in 1918. McGlennon also continued his political foray to be mayor of East Newark From 1907 to 1919.

In 1918, Democrat Cornelius McGlennon, the then majority leader of the New Jersey state senate, defeated republican William Ross by just 299 votes out of more than 25,000 cast. Cornelius was then the elected representative to the sixty-sixth Congress (March 4, 1919 - March 3, 1921), but was an unsuccessful candidate for reelection in 1920 to the sixty-seventh Congress. After this, he resumed law practice in East Newark, New Jersey. He however did serve as delegate to the Democratic National Convention in 1920. Cornelius was also elected to represent New Jersey's 8th District in the House of Representatives, serving from 1919 to 1921. In 1924, he was appointed judge of the court of errors and appeals and served until his death. Cornelius also was acting supervising principal at Harrison, New Jersey from 1926 to 1931. He died in Newark, New Jersey, June 13, 1931 at the age of 53. C. A. McGlennon I was interned in Holy Sepulchre Cemetery, East Orange, New Jersey. Find-a-Grave