Born in Paris, Cevert came from a complex family background. His father Charles Goldenberg was a Russian-Jewish émigré jeweller who had joined the French Resistance during the Nazi occupation to avoid deportation. Charles and his wife Huguette's children were registered under her surname, Cevert, rather than Goldenberg. Cevert's sister later married fellow Grand Prix driver Jean-Pierre Beltoise.
Cevert began his motorsport career aged sixteen on two wheels, racing his mother's Vespa scooter before graduating to his own Norton motorcycle. After completing national service he turned to cars, trained at the Winfield Racing School at Magny-Cours, and won the Volant Shell scholarship as the school's top student — a prize that provided him with an Alpine Formula Three car.
His 1967 Formula Three season in the Alpine was difficult, as he lacked the funds to maintain the car properly. After finding sponsorship and switching to a Tecno for 1968, he won the French Formula Three Championship, just ahead of Jean-Pierre Jabouille. That result brought him to the attention of Jackie Stewart, who struggled to pass Cevert in a Formula Two race at Crystal Palace in 1969 and told his team manager Ken Tyrrell to watch the young Frenchman. When Tyrrell needed a replacement driver three races into the 1970 season, that recommendation secured Cevert the seat.
Cevert made his Formula One debut at the 1970 Dutch Grand Prix at Zandvoort in a customer March-Ford run by Tyrrell. He scored his first championship point at the Italian Grand Prix at Monza. Over the following seasons Cevert became the devoted protégé of defending world champion Stewart, closing the gap to his senior teammate with almost every race.
The high point of his career came at the 1971 United States Grand Prix at Watkins Glen. Starting fifth, Cevert took the lead from Stewart on lap 14 as the Scot's tires faded. He held on despite late pressure from Jacky Ickx, whose Ferrari dropped out of contention when its alternator failed. Cevert crossed the line the winner, becoming only the second Frenchman to win a Formula One World Championship Grand Prix — Maurice Trintignant had been first, at Monaco in 1955 and 1958. Cevert received 50,000 US dollars in prize money. He finished the 1971 season third in the Drivers' Championship behind Stewart and Ronnie Peterson.
The 1972 season was more difficult. Cevert scored points only three times, with second places in Belgium and the United States. One bright spot came outside Formula One, where he finished second at the 24 Hours of Le Mans co-driving a Matra-Simca 670 with Howden Ganley. By 1973 he was again running at the front: he finished second six times, three times behind Stewart, and was visibly capable of running at the level of the reigning champion.
Stewart had already secured his third Drivers' Championship when the teams arrived at Watkins Glen for the 1973 season finale. He was secretly planning to retire after the race, and for 1974 Cevert had been confirmed as Tyrrell's new team leader. The accident that ended those plans occurred during Saturday morning qualifying.
In the fast uphill right-left combination called "The Esses," Cevert's Tyrrell 006 touched the left kerb and snapped right, striking the powder-blue safety barriers. The car spun back across the track and hit the outside barrier nearly head-on, uprooting the guardrail. Cevert died instantly of massive injuries. Stewart, who was among the first to reach the scene, withdrew from the session and announced his retirement from racing. The entire Tyrrell team withdrew from the race, meaning Stewart never started what would have been his hundredth and final Grand Prix.
Ronnie Peterson, who had been battling Cevert for pole position when the accident occurred, later told a Swedish television documentary that he had never seen anything like it, and described Cevert as his closest friend in Formula One.
Cevert received a significant documentary profile in the 1975 film The Quick and the Dead. His fatal crash at Watkins Glen, along with Helmut Koinigg's death at the same circuit a year later in 1974, prompted the addition of a chicane through the Esses complex in 1975 to slow cars through the sequence.
He remains one of the most admired drivers of the early 1970s — fast, stylish, and widely thought to be on the verge of becoming a genuine championship contender at the time of his death. Stewart later said publicly that Cevert was the most complete young driver he had ever seen. Cevert is buried in the Cimetière de Vaudelnay in Maine-et-Loire, France.