Although a top competitor in her own country, Davydova did not compete in international championships until the 1980 Summer Olympics, which were held in Moscow. The 1978 World Champion, Yelena Mukhina, had been paralysed after a fall from the balance beam, which left Davydova in a fight for the all-around title with reigning Olympic champion Nadia Comăneci of Romania and Maxi Gnauck of East Germany. The competition was a close affair, as Comăneci and Gnauck both missed the gold with only 0.075 points; with the Romanian's last exercise taking a long time to produce a note. In addition, Davydova won a gold medal with the team and a silver medal on the balance beam.
Davydova also participated in the 1981 World Championships, here second and last major international event. She finished third in the all-around final after an improper landing in the balance beam event, and won two more medals in the apparatus events.