Jensen was born in Annan, Scotland, to Margaret and Ivar Jensen, and is of Danish descent. She was raised by her mother, a special-needs teacher. She trained at Queen Margaret University, Edinburgh and was a member of the National Youth Theatre.
Jensen's first big television role was as Clare Donnelly, daughter of Glasgow criminal Jo-Jo Donnelly (played by Billy Connolly) in the 1993 BBC drama Down Among the Big Boys. In 1994, she played eccentric secretary Rosie McConnichy in the final series of BBC comedy May to December, as a replacement for the character of secretary Hilary (Rebecca Lacey), and Heather in Roughnecks, a BBC television series based around the workers on a North Sea Oil platform. Jensen had a small role in a Dangerfield episode "Contact" as the mother of a young girl with meningitis. She then appeared as Fiona Morris in EastEnders. She also appeared in the 2003 BBC drama Two Thousand Acres of Sky.
Jensen co-starred alongside Ricky Gervais in the BBC Two/HBO television programme Extras as the socially inept Maggie Jacobs. For her work on the first series, she received best television comedy actress and newcomer awards at the 2005 British Comedy Awards. In 2006, Jensen received two British Comedy Awards and a BAFTA nomination for her role in Extras. Her role in the 2007 Christmas Special earned her an Emmy Award nomination. In January 2006 she starred in the short-lived drama series Eleventh Hour on ITV. Later that year she played Steve Coogan's agent in A Cock and Bull Story. In 2007, she provided the narration for the film Taking Liberties.
In September 2006, Jensen's first major role on American television came when she took the role of regular character Christina McKinney on the ABC series Ugly Betty. Her character was originally an American, but when she met with the show's producers they liked Jensen's accent and changed the nationality of the character, keeping McKinney British. Production of Ugly Betty moved to New York from Los Angeles in mid-2008, causing a conflict for Jensen, who became a recurring member of the cast.
In January 2009, Jensen starred in No Holds Bard, a BBC Scotland one-off special comedy shown on Burns Night as part of a line-up of special programming to mark the 250th Birthday of Robert Burns. She is the narrator of Channel 4's Embarrassing Illnesses and Embarrassing Bodies. She also narrated an advertising campaign for Bank of Scotland and Persil detergent adverts for both television and radio. From September 2009 to May 2010, she appeared in the sitcom Accidentally on Purpose as Olivia. Jensen starred in the 2009 BBC comedy Nativity as Jennifer Lore, a Hollywood producer's secretary and the love interest of Paul Maddens, played by Martin Freeman.