He plays as a striker and got his breakthrough in 2008, scoring four goals in Allsvenskan alongside Henrik Larsson in Helsingborgs IF. Jönsson is commonly called "Sugröret" (The Straw) among his teammates. During the season 2010 in Allsvenskan, Helsingborgs IF Finished in second place just two points short of Malmö FF. Helsingborg however didn't allow this to let them down after a good season and finished it off on a high winning the Svenska Cupen in a final against Hammarby IF with Rasmus scoring the only goal in the game late on. On 29 August 2011, Helsingborgs IF confirmed that a deal had been agreed for Jönsson to join Bundesliga side VfL Wolfsburg on a four-year deal. Jönsson said that he was sad to leave Helsingborg but happy to take an important step forward in his career.
On 9 August 2013, it was announced that Jönsson would join the Danish Superliga club Aalborg Boldspilklub on a loan spell for the 2013/2014 season. On Sunday the 6th of October he scored twice against Viborg.