Jensen was born in Frederiksværk. He won the Oscar for Best Short Subject for his 1998 film Election Night. He received Oscar nominations in the live-action short category for his films Ernst & Lyset (1996) and Wolfgang (1997). He also wrote the script for After the Wedding which was nominated for an Oscar as Best Foreign film in 2007, The New Tenants, which won the 2009 Oscar for Best Live Action Short and In a Better World which won the Oscar for Best Foreign film in 2011 and the Golden Globe for Best Foreign film.
"As there was the Lubitsch touch and the Wilder touch, we can now speak of the Jensen touch -- a twinge of pathos acknowledged quietly, relying on our sympathy for the characters' bizarre frailties."