The son of four-star Admiral Joseph Strauss, Elliott Strauss followed his father into the naval profession. He graduated from the Naval Academy in 1923 and soon went on the shakedown cruise of the light cruiser USS Concord (CL-10). He subsequently was in the battleship USS Arkansas (BB-33), various destroyers, and the cruiser USS Nashville (CL-43)

he commanded the USS Brooks (DD-232). In the mid-1930s he was an assistant naval attaché in Great Britain, later flag lieutenant for Commander Atlantic Squadron, Rear Admiral Alfred Johnson. Strauss became a naval observer in England on the eve of World War II, then was the first U.S. naval officer on the staff of Lord Louis Mountbatten, Chief of Combined Operations. Strauss took part in the Dieppe operation and later served on various staffs in the months leading up to the invasion of France in June 1944. He later commanded the attack transport USS Charles Carroll (APA-28) and the cruiser USS Fresno (CL-121). After duty in OpNav he commanded Destroyer Flotilla Six and later served with the U.S. mission to the North Atlantic Treaty Organization (NATO). After retirement he was chief of the U.S. aid mission to Tunisia.