Svava Rós started her senior playing career in the summer of 2011 with Icelandic team Valur, making her debut in a match against Breiðablik, aged 15.

[4] During the 2018 Toppserien season, Svava Rós scored 14 goals in 21 matches, and was named in the Team of the Year.

[5] In November 2018, Svava Rós signed for Swedish team Kristianstads DFF for the 2019 Damallsvenskan season.

[6][7] In January 2021, Svava Rós joined French club FC Girondins de Bordeaux.

In January 2023, Svava Rós joined US National Women's Soccer League club NJ/NY Gotham FC.