Zavragin played for the youth club Shturm Chik, before joining Yugra Khanty-Mansiysk's amateur side in 2019. He made his professional debut for the club's MHL affiliate, Mamonty Yugry, during the 2021-22 season, making 31 appearances with a 2.13 goals against average, a .930 save percentage, and a 20-4-2 record during the regular season. He also made one postseason appearance, stopping seven of ten shots faced in one game. Returning to the team the following year, Zavragin's numbers dipped a bit, as he appeared in 21 games and worked to a GAA 2.49 with a SV% of .920.[1] He also debuted for Yugra at the VHL level that year, during which he would not allow a goal in a single appearance. Zavragin was productive for Yugry during their playoff run, finishing with a 2.68 GAA and a .927 SV% in 4 contests.
Prior to the start of the 2023-24 season, Zavragin was selected by the Philadelphia Flyers at 87th overall in the 2023 NHL Entry Draft.[1] 2023-24 was a breakout season for Zavragin, who started the campaign by fairly easily dispatching opposing offenses for Yugry. He finished the season with a GAA 1.63 and a .945 SV% in 9 games for them, then would later return to record a shutout for them during the MHL playoffs later on in the year. Zavragin would end up spending most of the year with Yugra though. There, he posted a similarly impressive 1.60 GAA and with a .943 SV% - the later the league's best mark for the season. Zavragin was also named VHL Best Rookie for his efforts. In the playoffs, he appeared in 8 games, finishing with a 2.11 GAA and a .933 SV%.
On May 2 2024, Zavragin was traded by Yugra to Kontinental Hockey League club SKA St. Petersburg for Yegor Gurzanov and Matvei Kabush, and immediately signed to a three-year contract.[2] In July, he was loaned to HK Sochi to start the season.[3] On October 9, after posting a .941 ave percentage in six games with Sochi, Zavragin was recalled from his loan by SKA.[4]