He became first among Soviets footballers who got transferred to West European world class club when he was picked by Juventus to replace the outgoing Michel Platini.

Zavarov is widely regarded to be among the greatest footballers in the history of the USSR and Ukraine, and in 2000 he was included in the Ukrainian Team of The Century according to a poll by the Ukrainsky Futbol weekly.

With the Zorya sports school team, twice Zavarov reached finals of a children republican tournament "Leather Ball" but, alas, never won it.

Zavarov made his debut at professional level on April 27, 1979 in away match against the defending Soviet champion FC Dinamo Tbilisi when he came on substitute for Igor Gamula after the first half.

[3] Eventually, Zavarov joined SKA Rostov-na-Donu (part of the Armed Forces society) and still has an ambiguous feeling about it whether he has done the right thing.

In 1981 with different coach SKA won the Soviet Cup and in the final game on assist of Zavarov, his "homeboy" Sergey Andreyev scored the winning goal against Spartak Moscow.

Two weeks after the cup final in the league match against Spartak Zavarov scored his first career hat-trick, however SKA playing at home lost to Muscovite team 3:4.

SKA Rostov-na-Donu leadership tricked Zavarov accused him in being "absent without leave" which would make him do another two years in disciplinary battalion.

[4] For almost 6 months he worked at the October Revolution Factory in Luhansk making train wheels until one of workers, two times Hero of Socialist Labor, blacksmith Kurylo didn't appeal to the Football Federation and disqualification was removed.

[4] After being reinstated following his disqualification Zavarov joined the factory team where he worked FC Zorya Luhansk that played in the Pervaya Liga (2nd tier).

At the end of summer of 1979, Zavarov took part in the 1979 FIFA World Youth Championship on the Soviet team losing in the final to Argentinians led by Diego Maradona.

[18] Because of his timid character, it was also argued that he lacked the necessary leadership skills to carry the team,[16] and fill the void left by Michel Platini in the advanced midfield playmaking role during the post-Trapattoni crisis.

He had a short spell as a head coach of FC Wil in 2003–04, however because he lacked the necessary UEFA licence, he was given the position of director of football with the club.