Zverev was a bronze medalist in the men's singles event at the 1983 Summer Universiade and won a singles gold medal at the Friendship Games, which were held in 1984. He did better than two years earlier at the 1985 Summer Universiade, winning both the singles gold medal and the doubles gold medal, partnering Sergi Leonyuk, with whom he was also a gold medalist in the 1986 Goodwill Games.