Rosen, a defenseman from Vaxjo, Sweden, began attracting attention from NHL teams in 2016-17, his third season with his hometown team in the Swedish Hockey League. The Toronto Maple Leafs signed him to a two-year contract on May 16, 2017.

Though Rosen was assigned to Toronto of the American Hockey League after training camp, he was quickly recalled and made his NHL debut against the New York Rangers on Oct. 7, 2017, and earned his first point, an assist, two nights later against the Chicago Blackhawks. But after playing four games with the Maple Leafs, he was turned to the AHL and spent the rest of the season there, helping Toronto win the Calder Cup championship.

Rosen spent most of 2018-19 in the AHL with Toronto, scoring seven goals and 46 points. The Maple Leafs recalled him again for four late-season games, and he scored his first NHL goal against the New York Islanders on April 1, 2019.

The Maple Leafs traded Rosen to the Colorado Avalanche on July 1, 2019. The Avalanche assigned him to Colorado of the AHL after training camp, and he split time between the Avalanche and the AHL before being reacquired by the Maple Leafs on Feb. 24, 2020.

Rosen signed a one-year, two-way contract with the Washington Capitals on July 2, 2025. His 34 points (10 goals, 24 assists) in the regular season and four points (two goals, two assists) helped Colorado of the AHL reach Game 7 of the 2025 Western Conference Final. He was traded to the St. Louis Blues from defenseman Corey Schueneman on Nov. 3, 2025.