Beganovic's parents, Fikret and Mirnesa, emigrated from Zenica, Bosnia and Herzegovina to Sweden before his birth. His father, a mechanic at Volvo, introduced him to Formula One, and Beganovic grew up watching the sport at home. The family spoke Bosnian at home and he frequently visited his grandmother in Zenica during childhood. He speaks Swedish, Bosnian, English, and Italian. His younger brother Emir plays football for Åtvidabergs FF.
Beganovic was introduced to karting through a national school program, after which his family purchased a used go-kart. He began karting competitively at age eight.
Beganovic won multiple national karting titles in Sweden. In 2018 he won the Swedish OK Junior category. In 2019 he won the OK category in both the Swedish and Italian championships and finished second in the WSK Euro Series. He joined the Ferrari Driver Academy in 2020 at the start of his single-seater career.
Beganovic made his single-seater debut in the 2020 Italian F4 Championship with Prema Powerteam. He took his maiden single-seater win at Imola, leading every lap from pole position in a weekend that included a complete podium sweep. He finished third in the overall standings with six podiums and 179 points. He also made select appearances in ADAC Formula 4 with Prema.
Beganovic entered FRECA with Prema in 2021, finishing thirteenth in the standings with 53 points, two rookie wins, and one podium. He also competed in the F3 Asian Championship that winter, finishing seventh overall with five rookie wins.
Beganovic remained with Prema for the 2022 FRECA season and dominated the campaign. He won from pole in the Monza opener and went on to take four victories — in Monza, Imola, Monaco, and Spa-Francorchamps — along with thirteen podiums and 300 points. He clinched the championship with a race to spare by finishing fourth at Mugello, winning the title by a commanding margin.
Beganovic graduated to the FIA Formula 3 Championship with Prema for 2023. He took four podiums across the season, including a second place in Monaco after qualifying on the front row. He finished sixth in the standings with 96 points, the highest-placed driver not to win a race.
Beganovic remained with Prema alongside Gabriele Mini and Arvid Lindblad. He claimed his maiden Formula 3 victory in the Melbourne feature race, overtaking both Mini and Leonardo Fornaroli for the lead. A second win came in mixed conditions in the Spa-Francorchamps sprint race. He again finished sixth in the standings, this time with two wins, one pole position, four podiums, and 109 points.
Beganovic stepped up to Formula 2 for the final two rounds of the 2024 season with DAMS Lucas Oil, replacing Juan Manuel Correa. On his debut in Qatar he qualified fourth and finished fifth in the feature race. At Yas Marina he finished on the podium after a sprint-race disqualification promoted him to third — his first F2 podium. He scored 22 points across his two-round cameo.
For 2025 Beganovic moved to Hitech TGR, partnering Luke Browning. He took his first Formula 2 pole position at Imola and scored his first F2 win in the Baku sprint race.
He returned to DAMS Lucas Oil for the 2026 season, taking pole position in Australia.
Beganovic made his Formula One free practice debut at the 2025 Bahrain Grand Prix, driving for Ferrari in place of Charles Leclerc. He finished fourteenth, second among the six rookies in the session. He drove for Ferrari again in the Austrian Grand Prix free practice session later that year and participated in the post-season young drivers' test at Yas Marina, completing 122 laps. In January 2025 he had his first F1 test at the Circuit de Barcelona-Catalunya in the Ferrari SF-23.
In July 2025 he sampled a Formula E car for the first time at the Tempelhof Airport Street Circuit in Berlin, driving for Mahindra Racing in the rookie test.