During the majority of his childhood, Daniel Wass played for Copenhagen-suburb-club BK Avarta, but was in 2006 scouted by Brøndby and promptly switched sides. After a year of playing for youth teams within the club, he joined the first team.
Wass made his debut for the Brøndby Danish Superliga team in the 2007–08 season by then-manager Tom Køhlert. He played half of the matches that season, but had to withdraw later on due to injury.
Following his recovery from injury Wass secured his place as right defender in the 2008–09 season playing in 28 games from a possible 33, helping Brøndby to reach third place in the Superliga, Brøndby's best position since their second place in the 2005–06 season.
Brøndby manager Kent Nielsen felt Wass should spend gain further developing at another club, so Brøndby chose to loan him out to Fredrikstad, a medal-competing side from Norway. On 21 August 2009, Wass made his debut in a 4–1 loss against Rosenborg and on 1 November 2009, Wass started and played the full 90 minutes scoring in a 5–0 win over Lyn. When the board of Fredrikstad FK signed a new manager, new coach Tom Nordlie informed Wass that he had no interest in an offensive right defender. As a result of that, Wass received no more than three games before returning to Brøndby in December 2009. For the remainder of the 2009–10 season, Wass played more games for Brøndby, securing his place in the starting eleven.
The season of 2010–11 proved to show Wass' importance for the Brøndby squad, when he was selected as first-choice right back for the first 19 games. In February, during the three-month winter break in the Superliga between December and March, Wass announced he would not be renewing his contract with Brøndby. From the beginning of January there had been rumours of Wass' summer transfer to Benfica. Wass did not comment on the rumours. In March 2011, with 14 matches of the season remaining, the attacking force of Wass was recognised, with Coach Henrik Jensen subsequently pushing the player up to right midfielder/winger. In his final matches for the club, Wass scored three goals, helping Brøndby to gain their third successive third place.
On 20 May 2011, Wass announced that he had signed a five-year contract with Benfica. However on 22 July 2011, he was loaned out to Evian Thonon Gaillard F.C. without taking part in any official games for Benfica. Upon moving to Evian, Wass joined Danish countrymen Stephan Andersen, Christian Poulsen and Thomas Kahlenberg. In the four months after joining Evian, Wass was kept out of the first team under Bernard Casoni, who preferred Brice Dja Djédjé, while Wass stated his intention to return to his parent club. On 15 October 2011, he made his debut, playing in right midfield, making an impact when he provided assist for Yannick Sagbo in a 2–1 loss against Saint-Étienne. Several weeks later, Wass scored in two consecutive games against Lorient and Rennes. He scored his third goal of the season in a 2–1 loss against Toulouse. Under new manager Pablo Correa, who succeeded Casoni, Wass secured a place in the first team, mostly being used as left-back, with Sidney Govou taking his former position in right midfield. After months of being kept out of the first team, he returned again, playing in left-back, due to Fabrice Ehret. Wass kept his place in the first team towards the end of the season when Guillaume Lacour took his position as left back, this time returning to the right midfield position. Two games before the end of the season, Wass scored his fourth goal in a 2–1 win over Ajaccio.
On 20 June 2012, Wass signed with Evian on a permanent basis for a €2.5 million transfer fee. On 12 August 2012, in the opening game of the season against Girondins de Bordeaux, Wass scored a free kick in a match that ended in a 2–3 loss for Evian. Like in the previous season, Wass played in various positions in defence and midfield. On 26 January 2013, Wass scored a further goal against Ajaccio in a 1–1 draw.
On 22 June 2015, Wass joined Celta on a four-year contract.
Wass is cousin of Nicki Bille Nielsen.