Andersson was born into a motorsport family in Sweden. Her father, Håkan, competed in the Swedish Rallycross Championship driving for Volvo and BMW, and her older sister Magda is a former FIA European Rallycross Championship event winner. As a member of the Sweden Junior National Team of Motorsport, Klara began her competitive career in karting before taking a five-year break and returning to rallycross in 2018.
After re-entering the sport, Andersson climbed the Swedish national rallycross ladder. She made a wildcard appearance at the 2019 RallyX Nordic season finale to build experience, then finished runner-up in the JSM (Junior) class of the 2020 Swedish Rallycross Championship, driving a BMW 120. The following season she stepped up to the SM (Senior) 2150 class and won the title outright in her first year at that level, competing against a field of more than 50 drivers — a result that significantly exceeded expectations.
Her performances attracted international attention through outings in the FIA RX2e Championship, where she recorded a best finish of fourth at Spa-Francorchamps. She also impressed at the Extreme E rookie test. In May 2022 it was announced that Andersson would join Niclas Grönholm at the Construction Equipment Dealer Team in RX1e — the top class of the FIA World Rallycross Championship — creating the first full-season gender-equal driver line-up in the championship's history.
At round 5 of the 2022 World RX season in Portugal, Andersson became the first woman in history to stand on a top-flight rallycross podium, a milestone in the sport's record books.
Andersson was originally signed by Xite Energy Racing to partner team owner Oliver Bennett for the 2022 Extreme E season. She contracted COVID-19 before the opening round in Saudi Arabia and was replaced by reserve driver Tamara Molinaro, who was subsequently retained by the team. Andersson stepped into the reserve role, then made her Extreme E debut in September 2022 at the Copper X-Prix in Chile. Standing in for Jutta Kleinschmidt — who had injured herself during free practice — at Abt Cupra XE, Andersson partnered Dakar Rally legend Nasser Al-Attiyah to a third-place finish, achieving a podium on debut.
For the 2024 season she joined the SUN Minimeal Team, though the campaign was curtailed when Extreme E cancelled rounds planned for Sardinia and Phoenix.
In September 2025, Andersson was confirmed as a driver for Carl Cox Motorsport's entry into the FIA Extreme H World Cup alongside Timo Scheider, who also serves as the team principal, continuing her progress across emerging off-road racing series.
Andersson's 2022 World RX podium in Portugal stands as a landmark moment for women in motorsport, ending a record-long wait for female representation on a top-flight rallycross rostrum. Her trajectory — from junior karting through national championship success to world-level competition — reflects a methodical rise that mirrors the pathways of many elite rallycross drivers, regardless of gender.