Representing the Soviet Union at the Women's Chess Olympiads of 1957 (Emmen) and 1963 (Split), Zvorykina produced two sparkling, medal-winning performances. The first resulted in a board 2 score of 12/14, securing both individual and team gold and even overshadowing the accomplishment of board 1 compatriot Olga Rubtsova, the World Champion. Her second appearance saw her post a similarly impressive 5½/6 score, helping the team to another gold medal. However, as this was achieved from the position of team reserve, no individual medal was awarded.