Prior to the 1968 Olympics, Besson, a virtual unknown working as a physical education teacher, qualified for the 400 m event. In the final, Britain's Lillian Board - the favourite for the gold - was way ahead of the rest of the field with just 100 m to go. With an amazing last sprint, Besson then moved up from fifth place to beat Board on the finish line by a tenth of a second. Her winning time of 52.03 seconds was 1.8 seconds better than her personal best.
The next year, Besson came close to winning another international title at the European Championships. In the 400 m final, she crossed the line almost level with her team mate Nicole Duclos, both in the world record time of 51.7. However, Duclos was awarded the victory after examination of the photo finish. Besson and Duclos also set a new world record in the 4 x 400 m women's relay in the same championships. In the 4 x 400 women's relay final, Besson, who was anchoring the French 4 x 400 m relay team, passed the finish line at the same moment as Lillian Board. Again, photo finish evidence determined Besson had come second.
After 1969, Besson would not win any more international medals. She qualified for the 1972 Summer Olympics in Munich, but was eliminated in the preliminaries, although she took fourth place in the relay. She retired from athletics in 1977.