Winkler won world championships in show jumping in 1954 and 1955. At the 1956 Olympic Games, in which the equestrian events were held in Stockholm, Sweden, because of quarantine restrictions in the main Games site at Melbourne, Australia, Winkler won gold medals in both individual show jumping and team show jumping. He then won team gold medals in show jumping at the 1960 Olympic Games in Rome and the 1964 Games in Tokyo. He led the German team to a bronze medal at the 1968 Olympics in Mexico City and a gold medal at the 1972 Olympics in Munich and finished his illustrious career with a team silver at the 1976 Olympic Games in Montreal.

Riding the same mare, Halla, for much of his early career, Winkler was the only horseman to have ridden all three legs of the team show-jumping course for Germany during their winning streak from 1956–1964. In the 1956 individual competition, Winkler sustained a severe groin injury and was almost unable to move, but his mare—without direction from her rider—completed the course flawlessly in the final round, bringing Winkler the gold.