Rasmussen is best known for his 1978 album Sweets to the Sweet.
Sweets To the Sweet was re-released in 2001 on Danish label Music Mecca.
He worked with, among others, Teddy Wilson, Ralph Sutton Horace Parlan, Ben Webster, Dexter Gordon, Oliver Nelson, Tom Waits, Al Grey, Wild Bill Davison, Harry Sweets Edison, Doug Raney, Ole Kock Hansen, Svend Asmussen, and Jesper Thilo.
He was one of the most sought-after bass players in the world and performed on 800 albums.
Rasmussen was the acoustic bassist for Norbert Susemihl’s New Orleans All Stars and for his own band, Hugo Rasmussen AllStarz which he formed in 1999 with Jakob Dinesen (tenor sax), Kasper Trandberg (cornet), Andrew Hyhne (trombone), Heine Hansen (piano) and Kresten Osgood (drums).
In 2009 he was awarded the title Ærespris (Honour-Prize in English) (Danish Music Award Jazz 2009) by the International Federation of the Phonographic Industry.