Dalmas began in French Formula Renault, finishing third in 1983 before winning the championship in 1984 with six victories from 12 races. He stepped up to French Formula Three in 1985, finishing second, then dominated the 1986 season to take the title ahead of Jean Alesi. He made his International Formula 3000 debut late in 1986 and contested the 1987 season with Oreca Motorsport, winning at Pau and Jarama to finish fifth in the standings.
Dalmas joined Larrousse for the final three races of the 1987 Formula One season. He outqualified teammate Philippe Alliot on his debut at the Mexican Grand Prix and finished fifth at the season finale in Australia, though the result carried no championship points as Larrousse had only entered one car for the full season.
He remained at Larrousse for 1988 but narrowly missed points twice — finishing seventh at Monaco and seventh again at Detroit — before missing the final two rounds after being diagnosed with Legionellosis. The 1989 season proved difficult: he qualified for only one of the first six races before moving to AGS, where he was eliminated in pre-qualifying in each of the final nine rounds alongside teammate Gabriele Tarquini. In 1990, Dalmas qualified for five of the 16 races, recording a best result of ninth in Spain.
In 1994 Dalmas returned to Formula One for two races with Larrousse, narrowly outqualifying teammate Erik Comas at Monza before retiring, then finishing 14th in Portugal in what proved his final Grand Prix.
Dalmas joined the Peugeot Sport works programme in 1991 alongside Keke Rosberg. In the World Sportscar Championship's final year of 1992 he partnered Derek Warwick, and the pair, along with Mark Blundell, won the 24 Hours of Le Mans from a dominant position despite a brief ignition failure overnight. They also won the World Sportscar Championship title that season.
His 1994 Le Mans victory came in the GT1-class Dauer 962 Le Mans Porsche alongside Hurley Haywood and Mauro Baldi. The team benefited from a larger fuel capacity advantage but were briefly compromised when Dalmas ran out of fuel on pit entry and had to be pushed to his box, and he later nursed the car home with a fragile driveshaft.
In 1995 Dalmas drove a McLaren F1 GTR at Le Mans alongside JJ Lehto and Masanori Sekiya, winning despite a late charge from Bob Wollek — his third Le Mans victory. His fourth came in 1999 at the wheel of a BMW alongside Pierluigi Martini and Joachim Winkelhock, the trio benefiting from a late puncture that ended the leading Toyota's challenge. A notable distinction across all four victories is that Dalmas never shared a car with the same co-driver twice across his twelve Le Mans appearances.
Dalmas competed in the FIA GT Championship from 1997, winning the 12 Hours of Sebring that year in a Ferrari 333 SP. In 1998 he partnered Allan McNish in the GT Championship, recording six consecutive podiums and finishing third in the standings. He continued competing at Le Mans through 2002, his final professional race coming in an Audi R8 for Team Goh where he finished seventh overall.
From the early 2010s Dalmas took on the role of driving advisor and safety car driver in the FIA World Endurance Championship, testing LMP1 machinery including the Porsche 919 Hybrid and Toyota TS050. He was named Grand Marshal for the 2025 24 Hours of Le Mans and stepped back from his advisory and safety car duties at the close of the 2025 WEC season.