He made his professional and Liiga debut with Kärpät near the conclusion of the regular season, allowing four goals in 33 shots of a shutout defeat to HPK on 8 March 2018.
[2] He returned to the Jr. A to be named as the league's best goalie, a first-team all-star and won a silver medal after the club finished in second place.
[3] In order to continue his development, Annunen was loaned by Kärpät to Mestis club Kokkolan Hermes, to begin the 2018–19 season on 31 July 2018.
On 6 May 2021, Annunen was reassigned by the Avalanche to begin his North American career with AHL affiliate, the Colorado Eagles, for the remainder of their season.
Adapting to the North American rink and style, Annunen was winless in his first 8 career AHL contests before earning a 5–4 overtime victory against the San Jose Barracuda on 29 October 2021.
He joined the Avalanche in time for the second period, assuming the backup duties for the remainder of an 8–2 defeat to the Toronto Maple Leafs.
On 30 November 2024, Annunen's tenure with the Avalanche ended after he was traded along with a sixth-round selection in the 2025 NHL entry draft to divisional rivals the Nashville Predators in exchange for Scott Wedgewood.
Annunen represented his country at the 2020 World Junior Championships, where he finished the tournament with a .916 save percentage and a 2.65 goals-against average in six contests.