Poehling was chosen by the Montreal Canadiens in the first round (No. 25) of the 2017 NHL Draft. He played for the United States at the 2018 and 2019 IIHF World Junior Championship and was named most valuable player and best forward at the 2019 tournament after his eight points (five goals, three assists) in seven games helped the U.S. win a silver medal. He also helped his country win gold at the 2017 IIHF U-18 World Championship and played at the 2016 Ivan Hlinka Memorial Cup.

The Canadiens traded Poehling to the Pittsburgh Penguins on July 16, 2022. After one season with the Penguins, Poehling signed a one-year, $1.4 million contract with the Philadelphia Flyers on July 1, 2023, and signed a two-year deal Jan. 27, 2024. He had 95 points (43 goals, 52 assists) in 283 NHL games when he was traded to the Anaheim Ducks on June 23, 2025. In two seasons with the Flyers, Poehling led forwards in blocked shots (180), and ranked second in face-off wins (836), face-offs (1,675), short-handed time on ice (235:17) and short-handed time on ice per game (1:37).

Poehling signed a four-year contract with the Ducks on March 5, 2026.