Zavarov, a product of Kirovograd's soccer school enjoyed a stellar 6 year career wearing a Dynamo Kiev's #9 jersey. He played both as a midfielder and a forward. On Dynamo Kiev's roster he won 2 Soviet Championships (1985, 1986), 2 Soviet Cups (1985, 1987), a UEFA Cup Winners' Cup (1985). He managed to beat out Igor Belanov (a 1986 Footballer of the Year) to become a 1986 USSR Player of the Year. Alexander was a member of Soviet National teams in 1986, and 1990 World Cups and was on USSR roster at 1988 European Championships, where he and his teammates finished second only to a magnificent Netherlands team..