Kleven signed a two-year, $3.2 million contract the Ottawa Senators on June 2, 2025, an average annual value of $1.6 million. He established himself as a regular in the lineup that season with 10 points (four goals, six assists) in 79 games while averaging 14:28 of ice time to help the Senators reach the Stanley Cup Playoffs for the first time since 2017, a six-game loss to the Toronto Maple Leafs in the Eastern Conference First Round.

Kleven, who scored his first NHL goal in his 39th game, Ottawa's 3-2 win at the San Jose Sharks on Nov. 27, 2024, made the Senators out of training camp. The second-round pick (No. 44) in the 2020 NHL Draft signed his entry-level contract March 23, 2023, after playing at the University of North Dakota, where he was teammates with future Senators Shane Pinto (2020-21), Jacob-Bernard Docker (2020-21) and Jake Sanderson (2020-22).

Kleven played for the United States at the IIHF World Junior Championship in 2021 and 2022, winning gold in 2021.