In the late 1970s Elgh was regarded in Sweden as an exceptional talent and was cited by Swedish motorsport journalists as a potential successor to Gunnar Nilsson and Ronnie Peterson, both of whom had reached Formula One from Scandinavia. He was managed throughout the 1970s and 1980s by former racing driver Torsten Palm.
Elgh began his single-seater progression through Formula Super Vee before moving to England in 1977 to race in the British Formula Three Championship with the Chevron Racing Team. He finished joint second in the BP Super Visco British F3 series and joint fifth in the Vandervell series that season — strong performances in a championship regarded as a direct stepping stone to Formula One.
In 1978 he stepped up to Formula 2 with the Fred Opert Racing team. His first season was competitive for a newcomer: a near-miss behind Bruno Giacomelli in Pau, with sixth-place finishes at Rouen-les-Essarts and Hockenheim, left him eleventh in the final standings. For 1979 Elgh moved to Tiga Race Cars and won his first F2 race at Enna-Pergusa.
His best European season came in 1981, when he drove for MM Mampe Team. He won at Vallelunga, finished second at Thruxton (behind teammate Roberto Guerrero) and at the Nordschleife (behind Thierry Boutsen), and took third at Spa, ultimately finishing third in the European Formula Two Championship.
Despite a McLaren test drive and three years of consistent F2 results, a Formula One seat did not materialise.
Elgh redirected his career to Japan in the early 1980s, a period when several European drivers were reshaping Japanese motorsport. He raced in Japanese Formula 2 and Formula 3000 and competed in the Japanese Endurance and Sports-Prototype Championships during the series' most competitive era of 1984–1988. He was a works driver with Dome Motorsport and later associated with Tom's Toyota, driving the 91C-V and 92C-V machinery.
In 1984 he raced for the Volvo team of compatriot Robert Kvist in the European Touring Car Championship, finishing second with Ulf Granberg in the 500 km race at Mugello.
His first Japanese long-distance win came in 1985, shared with Geoff Lees in the 500 km race at Suzuka, a round of the All Japan Sports Prototype Championship, driving a Dome 84C. The partnership with Lees proved productive: their last joint victory was the overall win in the 500 km race at Sugo in 1991, this time in a Toyota 91C-V. Elgh recorded four wins in the Japanese Sports Car Championship in total. He was regarded as a strong development driver and a loyal team player, qualities that kept him closely tied to Toyota even during periods of lean results. His active Japanese career concluded in 1995.
In 2006 Elgh entered the Dakar Rally, sharing a Motortecnica MAN 4x4 truck with Paolo Barilla and Matteo Marzotto. He returned for the 2009 Dakar Rally, where a severe accident saw his Nissan Patrol GR Y61 fall down a 100-metre cliff. The vehicle was destroyed but Elgh and co-driver Paolo Barilla escaped without serious injury.
In later years Elgh competed at the Monaco Historique Grand Prix. In 2012 he raced a 1977 F3 Chevron B38-Toyota — the same model he had driven in the 1977 British F3 season — and returned in 2016 at the wheel of a 1971 March 711-Cosworth Formula One car.
Elgh moved into motorsport broadcasting after his active driving years, working as an expert Formula One commentator in Sweden alongside Janne Blomqvist. The pair first worked together for TV4 and then continued when Viasat Motor acquired the Swedish Formula One broadcasting rights.
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