After beginning his career with Danish team Herning Blue Fox at the junior and men's level, Jensen moved to North America in 2010 to play with the Oshawa Generals.

Named to the OHL All-Rookie Team following his first year with the club, he was ranked 21st among North American skaters eligible for the 2011 NHL Entry Draft, where he was selected by the Vancouver Canucks 29th overall.

Jensen spent most of the 2012–13 season with AIK of the Swedish Elitserien before returning to North America and playing for Chicago, the AHL affiliate of the Canucks.

Jensen made his NHL debut with Vancouver in 2013 and appeared in 24 games for the Canucks over three seasons, spending most of his time with the team's AHL affiliates, the Chicago Wolves and Utica Comets, before being traded to the New York Rangers in 2016.

Playing with the Oshawa Generals, his paternal grandparents Poul and Utta Jensen, as well as two uncles live nearby Richmond Hill, Ontario.

[8] That same month, he competed in the 2011 CHL Top Prospects Game for Team Cherry, alongside Generals teammates Boone Jenner and Lucas Lessio.

Following his draft, he characterized himself as "a power forward and goal scorer who protects the puck well", while the Canucks assistant general manager Laurence Gilman highlighted the quick release of his shot.

[1] In becoming a part of the Canucks organization, Jensen joined countryman Jannik Hansen, who led a wave of Danish players beginning to play in the NHL.

After the Generals were eliminated in the first round of the 2012 OHL playoffs, Jensen was assigned to the Canucks' American Hockey League affiliate, the Chicago Wolves.

[28] On 8 January 2016, while with AHL affiliate, the Utica Comets, Jensen was traded by the Canucks along with a 6th round draft pick in 2017 to the New York Rangers in exchange for forward Emerson Etem.

[31] As a restricted free agent from the Rangers, Jensen opted to pause his NHL aspirations in agreeing to an optional two-year contract with Jokerit on 12 July 2017.

As a free agent in the off-season, Jensen moved to the Swiss National League, signing a two-year contract with SC Rapperswil-Jona Lakers on 6 May 2022.

[notes 2] Seven months later, Jensen joined Denmark's under-20 team, competing in Division I of the 2010 World Junior Championships, held in Megeve and Saint-Gervais, France.

[44] Finishing with a team-leading 15 points over 5 games, including a pool-leading 13 goals,[45] Jensen was distinguished as Denmark's best player and the pool's best forward by the coaches and the directorate, respectively.

Jensen was chosen to the team as the lone player competing with a North American club at the time (the Oshawa Generals)[52] and was also named an alternate captain.

[notes 4][53] During the preliminary round, Jensen was named Denmark's player of the game after recording an assist and five shots on goal in a 10–2 loss to Canada.

[22] Jensen would make his debut at the World Championships in 2013 when he was added to Denmark's roster following the elimination of the Vancouver Canucks from the Stanley Cup playoffs.