At the start of his cycling career Edy Schütz finished third at the 1961 Luxembourg amateur road championship, followed by second in both 1963 and 1964. In 1964 Schütz also won the overall classification at the Tour of Austria and competed in the road race at the Tokyo Olympics. After the Games he turned professional, winning five consecutive Luxembourg road race titles from 1966 to 1971. In the same year as his first national title Schütz also made his début at the Tour de France, winning the 18th stage of the race. He then rode in the next five editions of the Tour de France, along with three editions of the Giro d’Italia (1967–69) and two editions of the Vuelta a España (1967, 1971). After retiring from competitive racing Schütz ran a bicycle shop in Dudelange. His father-in-law Jacques Majerus was also a cyclist, having rode at the 1936 Berlin Olympics.