Zub played as a youth with Amurskie Tigry Khabarovsk, the junior affiliate to Amur Khabarovsk.[1] He made his professional debut with Amur Khabarovsk in the Kontinental Hockey League (KHL) during the 2014–15 season.[2] He was named the KHL Rookie of the Month of October during the 2015–16 season.[1][3] During the 2016–17 season, his third with Khabarovsk, Zub contributed with 9 points in 32 games before he was traded to a powerhouse club SKA Saint Petersburg on 1 December 2016.[4] Zub maintained a regular role on the SKA blueline to close out the season. In the post-season, he helped the club claim the Gagarin Cup.[5] In his final season with SKA Saint Petersburg, Zub set career highs in the KHL with 13 goals and 22 points in 57 games.[6]
Zub's play in the KHL and the Olympics attracted the attention of the National Hockey League (NHL)'s Ottawa Senators. On 1 May 2020, the Senators signed him to an NHL-standard entry-level contract as an undrafted free agent for one season.[7] He scored his first NHL goal in a 6–5 overtime win against the Toronto Maple Leafs on 15 February 2021 at the Scotiabank Arena in Toronto.[8] In his first season, he was regularly paired on defence with Ottawa's top defenceman Thomas Chabot.[9] Upon joining the Senators he quickly solidified himself in the team's defence, seen as the Senators' best defensive defenceman.[10]
On 14 May 2021, Zub signed a two-year, $5 million extension with the Senators.[11] The fans in attendance at the Canadian Tire Centre took to Zub, and called out "Zuub" when he played the puck. Zub, due to his limited English had to be reassured by his teammates that he was not being booed, and came to enjoy it.[12] During the 2022–23 season, Zub struggled with injuries. On 27 October suffered and upper body injury that kept him out of the lineup until 18 November.[13] On 2 December Zub suffered a broken jaw after being struck in the face by a puck in a 3–2 win over the New York Rangers.[14] On 21 December 2022, Zub was signed to a four-year, $18.4 million extension by the Senators.[5][15] Zub returned to the lineup on 30 December versus the Detroit Red Wings.[16] Then on 14 January 2023, Zub sustained a lower body injury versus the Colorado Avalanche after blocking a shot.[17] He returned to play in the 11 February 2023 game versus the Edmonton Oilers.[18]