Born in Waiblingen on 6 October 1951, Winkelhock was the older brother of Joachim and Thomas, both of whom also competed in motorsport. He began racing in Formula Two in 1978 and survived a serious accident at the Nürburgring Nordschleife early in his career. The front wing of his March 802 was bent early in the race, reducing downforce; on the fourth lap, the car became airborne over the crest of Quiddelbacher Höhe before the Flugplatz corner, somersaulting backwards and then sideways. Winkelhock emerged without fatal injury.
He gained wider attention in 1979 when he co-drove a BMW M1 Group 4 at the 24 Hours of Le Mans alongside Hervé Poulain and Marcel Mignot. The car had been painted by pop artist Andy Warhol as part of BMW's Art Car series. The trio finished sixth overall and second in class.
Winkelhock's first Formula One attempt came as a stand-in for the injured Jochen Mass at Arrows in Italy. He secured a full drive with ATS in 1982, a team that had BMW as its engine supplier from 1983 onwards. The BMW turbocharged engine gave him competitive qualifying pace on several occasions in 1983 and 1984, but the car suffered persistent reliability problems, leaving him with more retirements than results. His best Grand Prix finish was fifth at the 1982 Brazilian Grand Prix.
At the time of his death, Winkelhock was racing for the Skoal Bandit-sponsored RAM Racing team in Formula One. The 1985 season had been difficult, with a best finish of 12th at the French Grand Prix at Paul Ricard. RAM was already struggling financially and folded before the end of the season after Winkelhock's death.
Alongside his Formula One commitments, Winkelhock maintained an active sportscar and touring car programme. He won the 1000 km Monza in 1985 partnered by Marc Surer.
On 11 August 1985, Winkelhock was driving a Porsche 962C for Kremer Racing, co-driven by Surer, in the Budweiser 1000 km World Endurance Championship event at Mosport Park near Bowmanville, Ontario, Canada. He crashed heavily into a concrete barrier at the circuit's fearsome Turn 2. Winkelhock was transported to Sunnybrook Medical Center in Toronto, where he died from his injuries the following day, 12 August 1985.
His death led to Winkelhock being replaced at RAM Racing by Northern Irish driver Kenny Acheson, though the team's financial difficulties meant it did not complete the season regardless.
Winkelhock never won a Formula One race, but his career spanned two of the sport's most competitive eras and he was regarded as a determined, quick driver hampered by rarely competitive machinery. His connection to the BMW Art Car programme at Le Mans in 1979 gave him a moment of cultural as well as sporting significance. His family's legacy in motorsport continued through his son Markus — who led the 2007 European Grand Prix and won the FIA GT1 World Championship in 2012 — and his brothers Joachim and Thomas, who also raced at the top level.