Falk started his career at Middelfart G&BK and moved to Odense Boldklub (OB) alongside future Tottenham Hotspur and Inter Milan player, Christian Eriksen in 2005.

On 20 January 2016, Falk signed a pre-contract with Copenhagen, as he turned down a contract extension for OB, meaning that he could move as a free agent in the summer.

Falk made his Copenhagen debut on 13 July 2016 and scored the final goal in the 3–0 Champions League qualification away win over Crusaders.

[3] On 5 August 2020, Falk scored the third goal in a 3–0 win over Turkish champions İstanbul Başakşehir to secure a spot in the quarter-finals of the Europa League.

[5] He played the first half of a 2–1 FIFA World Cup qualification win over Malta on 6 September 2013.