In 2013-14, Soshnikov had 13 points (five goals, eight assists) in 10 games for Atlant Moscow's junior team before moving up to play for Atlant in the Kontinental Hockey League, where he had five points (two goals, three assists) in 33 games. The following season, Soshnikov had 32 points (14 goals, 18 assists) in 57 games and was signed as a free agent to a three-year, entry-level contract by the Toronto Maple Leafs on March 20, 2015.
Soshnikov spent the majority of the 2015-16 season with Toronto of the American Hockey League, where he had 28 points (18 goals, 10 assists) in 52 games. He made his NHL debut in a 2-1 loss to the Tampa Bay Lightning on Feb. 29, 2016 and scored his first NHL goal in the next game, a 3-2 loss to the Washington Capitals on March 2. Soshnikov finished the season with five points (two goals, three assists) in 11 games.