Kjellbin grew up in Mölndal, in Göteborg, and began skating at age four. As a child, she participated in various sports and games, including golf, swimming, football, and chess. From 2008 to 2010, she split her time between the Hanhals IF women's team and the U16 boys' team.[2] In her rookie Riksserien season, 2009–10, she picked up five assists in 28 games as Hanhals finished second to last.
In 2010, she signed with Linköping HC Dam, choosing the club over an offer from HV71 Dam. She won the Riksserien championship twice with the club in 2014 and 2015. In the 2015–16 season, she scored a career-high 21 points in 36 games. She scored 6 points in 36 games in the 2018–19 season and, before the beginning of the playoffs, announced that she would take a break from hockey for personal reasons.[3] She later revealed that she had sustained concussions in two different games, which caused
She joined HV71 in 2019, after nine seasons with Linköping, seeking a change in environment and because her job outside of hockey was based in Jönköping.[4] She scored 11 points in 36 games in her first season with HV71, as the club finished in first place during the regular season and made it to the playoff finals against Luleå HF/MSSK before the season was canceled due to the COVID-19 pandemic in Sweden. She criticized the league's decision not to award the championship to any team, despite HV71 having won the first and only completed match of the finals, stating that: "Considering the season we had, it feels very empty."[5] She returned to HV71 for the 2020–21 season and was the team's third highest scoring defenseman in the regular season.
^Hedlund, Robert (1 June 2021). "Välkommen Anna Kjellbin!". Luleå Hockey (in Swedish). Archived from the original on 24 June 2021. Retrieved 17 June 2021.
