Despite the career of cyclist Rudolf Rasmussen overlapping with World War II, he still managed to win several amateur races in his native Denmark. In 1939 Rasmussen won the Nordic Championship road race in København before he became the Danish amateur road race champion one year later. He won another Danish amateur title in 1942, also finishing second in 1943 and 1944, and third in 1945. After the war had ended Rasmussen was able to compete in races outside of Denmark, finishing fourth overall at the 1947 edition of the Tour of Sweden. One year later he competed in the road race at the 1948 London Olympics but did not finish the course. His brother Jørgen participated in the road race at the 1952 Helsinki Olympics.