Born Albert François Cevert in Paris, he was the son of Charles Goldenberg, a Parisian jeweller of Russian-Jewish descent who had joined the French Resistance during the Nazi occupation, and Huguette Cevert. The family children were registered under their mother's surname to avoid drawing attention during the occupation. Cevert's sister would later marry fellow Grand Prix driver Jean-Pierre Beltoise.
Cevert began motorsport on two wheels at age 16, progressing from his mother's Vespa to his own Norton motorcycle. After completing national service he turned to cars, attending the Winfield Racing School at Magny-Cours, where he won the Volant Shell scholarship as the top student, earning a prize Alpine Formula Three car. His early Formula Three campaign was difficult for lack of funds, but after securing sponsorship for 1968 he switched to a Tecno and won the French Formula Three Championship.
He joined the works Tecno Formula Two team in 1969 and finished third in that championship, also driving in the Formula Two class of the German Grand Prix. It was during an F2 race at Crystal Palace that year that Jackie Stewart struggled to find a way past the young Frenchman. Stewart told Ken Tyrrell to watch him closely; when Tyrrell needed a replacement driver at short notice in 1970, he recalled that recommendation.
Cevert replaced Johnny Servoz-Gavin, who had suddenly retired three races into the 1970 season. He debuted at the Dutch Grand Prix at Zandvoort in a customer March-Ford, earning his first championship point with sixth place at Monza.
In 1971, with the Tyrrell team now constructing its own cars, Cevert was consistently fast. He finished second in France and Germany, both times behind Stewart. The season ended in triumph at Watkins Glen, where he took the lead from Stewart on lap 14 as the Scot's tyres faded in intense heat. Despite a late surge from Jacky Ickx — who had set the fastest lap and closed to within 2.2 seconds before his Ferrari's alternator failed — Cevert held on to win by a comfortable margin. It made him only the second Frenchman to win a Formula One World Championship Grand Prix, after Maurice Trintignant, and earned him third in the Drivers' Championship behind Stewart and Ronnie Peterson.
The 1972 season was a disappointment: Cevert scored points only three times, with second places in Belgium and the United States and fourth in France. A highlight outside Formula One was his second-place finish at the 24 Hours of Le Mans co-driving a Matra-Simca 670 with Howden Ganley.
By 1973 Cevert had developed into a driver capable of matching Stewart at almost every circuit. He finished second six times during the season, three of those behind his teammate. As the year drew to a close, Stewart had already arranged to retire after the final race in the United States; for 1974, Cevert was to have become Tyrrell's team leader.
At the 1973 United States Grand Prix, Stewart had already secured his third World Championship. On Saturday morning qualifying, Cevert was pursuing a first pole position when he entered the fast right-left uphill Esses complex at Watkins Glen. His car brushed the kerb on the left side, snapping across the track at approximately 240 km/h and striking the powder-blue guardrail on the right at near 90 degrees. The impact uprooted the barrier, and the car was destroyed. Cevert died instantly from massive injuries. He was 29 years old.
Stewart, one of the last at the scene, found the accident so devastating that Tyrrell immediately withdrew both entries for the race. Stewart never competed again. Ronnie Peterson described the sight as the worst he had ever witnessed, and remained visibly shaken when discussing it years later.
In response to the accident, a chicane was installed at the Esses in 1975. That modification was removed in 1985 after the circuit lost its Formula One race in 1981.
Cevert is buried in the Cimetière de Vaudelnay in the village of Vaudelnay, Maine-et-Loire. His 1971 win at Watkins Glen remains a milestone in French motorsport history. He was extensively profiled in the 1975 Formula One documentary The Quick and the Dead. The 2013 film Rush portrays a composite of his fatal accident merged with elements of a later incident at the same circuit. A film documentary shot just minutes before the fatal qualifying session captured Stewart and Cevert in a lively debate about gearing technique through the Esses — a detail that underlines how the crash cost the sport one of its most complete and promising talents at the precise moment he was ready to lead.