Leading up to the
2015 NHL Entry Draft, Leskinen was ranked 114th overall International Skaters by the
NHL Central Scouting Bureau. While he went undrafted in North America, Leskinen signed a three-year contract extension with KalPa on 29 April 2015. He signed a two-year extension with KalPa on 21 November 2017. In June 2018, Leskinen was invited to attend the
Montreal Canadiens of the
National Hockey League (NHL) training camp. In December 2018, Leskinen helped lead Kalpa to their first
Spengler Cup Championship. He recorded one point during the tournament, an assist off captain
Tommi Jokinen's goal in the semi-finals. Leskinen had a breakout season in the final year of his contract. He recorded a career-high 31 points, 23 assists, and 8 goals in 57 games during the
2018–19 season. As a result, Leskinen signed a two-year, entry-level contract with the Montreal Canadiens in May 2019. He was invited to their annual development camp in June that year and competed in their pre-season games. He made his NHL debut in a 3–2 defeat against the
Colorado Avalanche on 5 December 2019. After playing five games for the Canadiens, Leskinen returned to the
Laval Rocket for the remainder of the season. On 4 August 2020, Leskinen was returned by the Canadiens on loan to former club, KalPa of the Liiga, to begin the 2020–21 season due to the delayed North American season as a consequence of the
COVID-19 pandemic. As a free agent from the Canadiens organization, Leskinen was signed by Finnish outfit,
Jokerit of the KHL, agreeing to a two-year contract on 15 June 2021. After splitting time between the foregoing and
Tappara of
Liiga, Leskinen returned to the Canadiens organization via a one-year, two-way contract signed on 14 June 2022. Following the
2022–23 season within the Canadiens organization, Leskinen left the organization as a pending free agent and returned to his former club, Tappara of the Liiga, on a two-year contract on 10 May 2023. After completing his two-year stint with Tappara in the Liiga, Leskinen as a free agent was signed to a two-year contract with Swedish champions, Luleå HF of the SHL, beginning from the
2025–26 season on 22 May 2025. ==International play==