Jensen was born in Hillerød, Denmark, he began his youth career at Hillerød G&IF before going on to F.C. Nordsjælland where he has played since he was 13 years old, signing his first contract with the club at age 16. He would later go on trial with Celtic, but decided against a move to the Scottish club stating that though moving to a large foreign club would be a dream move, moving at 16 years of age was too young and that staying at FCN would be better for his development. David Jensen has an older brother Daniel Jensen, a former FCN defender who also played for FC Fredericia.
Senior career
FC Nordsjælland
David Jensen was promoted to reserve goalkeeper as backup to first choice Jesper Hansen in 2010. On 2 August 2011, he signed a new contract with FCN, keeping him at the club until 2014. On 31 August2011, Jensen made his senior debut in goal against hometown team Hillerød Fodbold in the second round of the Danish Cup, keeping a clean sheet with a 5–0 win. He would play his second match for the senior team in the following round of the Danish Cup on 21 September, in a 2–0 win over 1st Division side FC Hjørring. David Jensen would go on to make his Superliga debut on 24 September, coming on as a 44th-minute substitute for the injured Jesper Hansen, in a 1–0 loss to AGF. He would make his first league start on 2 October, in a 1–0 win over SønderjyskE, but had to come off at half time because of an injury to his knees from a collision with another player. He was expected to be out of action anywhere between four and six weeks. After not featuring regularly for Nordsjælland, Jensen was loaned to FC Fredericia and AB during the 2012/13 season. Upon returning to the club he would become the first choice keeper for the next three seasons.
Jensen transferred to Utrecht of the Eredivisie on 28 July 2016. He made his first appearance for the club as an injury substitution in a 3–3 draw against Feyenoord. In his first season at the club he helped Utrecht to a fourth place finish in the Eredivisie and qualification to the Europa League. In 2017, Jensen recorded his first clean sheet in European competition during the 2017–18 Europa League Second Qualifying Round in a 0–0 draw against Maltese club, Valletta.
On 29 January 2020, Jensen signed with the New York Red Bulls of Major League Soccer. Jensen started the club's first game of the 2020 season and earned the victory over FC Cincinnati, 3-2.
