Johansson was born and raised in Växjö, Sweden. He began in kart racing, winning the Swedish Championship in 1973, and progressed through Formula Ford — winning the Swedish title in both 1977 and 1979. He competed in the British Formula Three Championship from 1978 to 1980, winning the title in 1980 driving for Project Four, the team operated by future McLaren chief executive Ron Dennis.
Johansson made his Formula One debut at the 1980 Argentine Grand Prix with Shadow but failed to qualify for both races he entered that year. After spending 1982 in European Formula Two with Spirit Racing, he returned to Formula One with Spirit in 1983, making his first classified race starts and recording a best finish of seventh at the Dutch Grand Prix.
After brief outings for Tyrrell and Toleman in 1984 — where he recorded a fourth place at Monza in Toleman livery alongside future triple champion Ayrton Senna — Johansson stepped in at Ferrari mid-season 1985 after René Arnoux was dismissed. In only his second race for the team, the San Marino Grand Prix, he led for half a lap before running out of fuel near the finish, and subsequently finished second at the Canadian and Detroit Grands Prix. In 1986, Johansson frequently outpaced Ferrari teammate Michele Alboreto and finished fifth in the World Drivers' Championship — the best position of his Formula One career. Observers including Murray Walker and James Hunt noted that Ferrari may have been releasing the wrong driver when Johansson was replaced by Gerhard Berger for 1987.
Johansson moved to McLaren as number two to double champion Alain Prost for 1987. Though McLaren was less dominant that year, Johansson scored five podiums, including second in Belgium and finishes in Brazil, Spain, and Japan. He famously crossed the line at the German Grand Prix on three wheels after a late puncture. He finished sixth in the championship but was released at season's end as McLaren had secured Ayrton Senna for 1988.
Johansson joined Ligier for 1988 alongside René Arnoux, but the Judd-powered JS31 was uncompetitive, and he failed to qualify six times. He moved to the new Onyx team in 1989 and scored a popular third-place finish in Portugal — both his last podium and the team's only podium of its existence. After falling out with Onyx's new owner Peter Monteverdi in early 1990, he made further appearances for AGS and Footwork in 1991 before leaving Formula One. His record of twelve podiums without a win was later matched and surpassed by Nick Heidfeld.
Johansson transitioned to CART Championship Car racing in 1992, winning the Rookie of the Year award with two third-place finishes. He earned his first CART pole at Portland in 1993 but, as in Formula One, never converted a pole or strong run into a race win. He competed through 1996 in 73 starts, with his best season overall coming in 1994 when he finished eleventh. At the 1996 Molson Indy Toronto race, he was involved in an accident that claimed the lives of driver Jeff Krosnoff and marshal Gary Avrin.
Johansson entered fifteen editions of the 24 Hours of Le Mans between 1983 and 2012. In 1997 he co-drove the Joest Racing TWR-Porsche WSC-95 alongside Michele Alboreto and a young Tom Kristensen — Kristensen's first of a record nine Le Mans victories — to take the overall win. Earlier in the same year he also won the 12 Hours of Sebring sharing a Ferrari 333 SP with Andy Evans, Fermín Vélez, and Yannick Dalmas. He also claimed two World Sportscar Championship race wins in the 1980s, at Mugello in 1983 and Spa in 1988. In the 2000s he competed across the American Le Mans Series and established his own teams, including Johansson-Matthews Racing in 2000 and American Spirit Team Johansson in CART in 2003.
Johansson has managed several prominent drivers, including Scott Dixon, Felix Rosenqvist, Romain Grosjean, and Ed Jones. He is noted as a keen artist and watch designer, and has provided expert commentary for Viasat Motor's Formula One coverage. The song "Speedway at Nazareth" by Mark Knopfler was written in his honour.
Across eleven seasons in Formula One with ten different teams, Stefan Johansson was regarded as a consistently quick and professional driver whose twelve podiums without a victory reflected circumstance as much as capability. His 1997 Le Mans victory with Joest — the race that launched Tom Kristensen's legendary record — stands as the defining result of an endurance career conducted alongside his Grand Prix work and long into retirement.