Career
Metalurh Zaporizhya
Kryvtsov is a product of Metalurh Zaporizhya Youth school system, where he was trained by Viktor Tryhubov. He made his debut for the club at the age of 17 in a 1-1 draw with Chornomorets Odesa on 3 May 2008. His first goal for the club came in a 2-0 victory over Illichivets Mariupol on 27 February 2010.
He left the club in May 2012 to join Shakhtar Donetsk for an undisclosed fee.
Shakhtar Donetsk
On 11 May 2010 Serhiy, along with Taras Stepanenko, signed for Football Club Shakhtar Donetsk from Metalurh on a five-year deal. His first appearance for the club came on 10 November 2010 in a 1-0 Ukrainian Cup victory over his old club, Metalurh Zaporizhya.
He made a further two appearances in his first season, both coming in the league. The following season he made 6 appearances for the club, 5 of which came in the league, as Shakhtar clinched the Premier League title and the Ukrainian Cup.
He made his first appearance of the 2012-2013 season on 26 August in a 3-0 victory over Karpaty Lviv at the Donbass Arena.
He played the full 90 minutes, picking up a yellow card in the 85th minute. He also played in a 3-1 victory over Dynamo Kyiv as a 70th-minute substitute, replacing the injured Oleksandr Kucher who scored twice. He made his first appearance for the senior national side in a 4-0 friendly defeat to Czechoslovakian Republic on 6 September 2011, playing the full 90 minutes.