Nørgaard came to prominence with Brøndby and was part of the Denmark squad at UEFA Euro 2020.

A defensive midfielder, Nørgaard began his career in his native Denmark with BK Heimdal and Espergærde IF, before entering the youth system at Lyngby in 2008.

Two months later, Nørgaard transferred to Bundesliga club Hamburger SV and signed a three-and-a-half year contract.

On 21 August 2013, it was announced that Nørgaard had joined Danish Superliga club Brøndby on a four-year contract. He was named as Brøndby's Player of the Year for the 2017 calendar year and won the first silverware of his career with the 2017–18 Danish Cup. By the time of his departure in July 2018, he had made 145 appearances and scored 11 goals.

On 19 July 2018, Nørgaard moved to Italy to sign for Serie A club Fiorentina. He signed a four-year contract, but made just six appearances before departing the club in May 2019.

On 28 May 2019, Nørgaard moved to England to join Brentford on a four-year contract, with the option of an additional year, for an undisclosed fee. The move reunited him with his former Brøndby and Denmark youth coach Thomas Frank. He made 45 appearances during the 2019–20 season, which ended with defeat in the 2020 Championship play-off Final. In September 2020, he signed a new four-year contract, with the option of a further year.

He began the 2020–21 season as a virtual ever-present in midfield and winning senior international recognition with Denmark, but an ankle injury suffered during a match versus Preston North End on 4 October 2020 prevented him from making only one appearance during the following four months. Nørgaard returned to the team in late February 2021 and late in the season he was deployed as the sweeper in a 3-5-2 formation. An injury restricted Nørgaard to only one appearance during Brentford's 2021 playoff campaign, which culminated in promotion to the Premier League after a 2–0 victory over Swansea City in the Final. He finished the season with 22 appearances and one goal.

Nørgaard was named in the Denmark squad for Euro 2020. He appeared as a substitute in five of the six matches of the Danes' run to the semi-final.