Born in Binghamton, New York, Mitchell attended the Binghamton schools and had three years of college training at the American Law Institute and Columbia University, New York City. He moved to Evansville, Indiana, in September 1937.

He engaged as a warehouseman and later as district manager for a large food distributor 1934-1937. In 1937 Mitchell purchased a half interest in a food marketing and brokerage company and served as president. At age twenty-seven after purchasing part of the company he had accumulated a net worth of over one million dollars which with inflation would be around $15 million.

Following his success in the food service industry he served in the United States Navy from November 1942 until his discharge as a lieutenant commander in January 1946, having been commanding officer of Underwater Demolition Teams in the Pacific Theater for two years. He was awarded the Silver Star Medal at Okinawa following his honourable discharge.

Mitchell was elected as a Republican to the Eightieth Congress (January 3, 1947 – January 3, 1949). He was an unsuccessful candidate for reelection in 1948 to the Eighty-first Congress.

He served as delegate in 1952 and 1956 to Republican National Conventions. In March 1971 Mitchell, who was sixty at the time, assaulted Congressman Hale Boggs, 57, to the floor in the men's room of the Statler Hilton during a Congressional dinner; Boggs threatened suit; Mitchell replied he "would meet Boggs in court or in gym". The original argument was due to Congressman Mitchell's objection to the abusive cracks that Boggs was making about the Nixon Administration.

Mitchell resided in Evansville, Indiana, where he died December 11, 1979. He was interred in Sunset Memorial Park.

• United States Congress. "E. A. Mitchell (id: M000809)". Biographical Directory of the United States Congress.

  This article incorporates public domain material from the Biographical Directory of the United States Congress