Lauge began his handball career in
Bjerringbro FH. In 2009 he signed a 3-year deal with
Bjerringbro-Silkeborg. He signed a 3-year contract with
THW Kiel on 28 February 2013, which began with the start of the 2013–2014 season. On 5 May 2015 Lauge signed a 3-year deal with
SG Flensburg-Handewitt. On 5 March 2018 it was announced that Rasmus Lauge would continue his career with Hungarian side
Telekom Veszprém joining the club from the 2019/2020 season. He signed a two-year contract with the club. In 2023 he returned to Bjerringbro-Silkeborg. ==Honours==