The heir to a considerable family transport business, Hesnault only began racing after completing his military service in the French parachute corps, which included action in West Africa. He made his racing debut in the French Formula Renault championship in 1980, then moved to Formula Three two years later.
In 1982, Hesnault began the season driving a Martini MK37-Alfa Romeo and was the leading national series runner in the Magny-Cours European F3 Championship race. Outpaced by Ralt RT3 rivals, he used his UFO/Antar sponsorship to switch to an ex-Roberto Moreno Ralt RT3 in September and immediately won at Albi. He finished third in the French F3 championship that year, behind Pierre Petit and Michel Ferté, and won twice overall.
For 1983, Hesnault joined Dave Price Racing and drove a Ralt RT30-Volkswagen. He improved steadily across the season, finishing second in a European round at Magny-Cours and seventh at Monaco, and went on to win five times. He ultimately finished just two points behind French Champion Michel Ferté in the championship standings. The 1983 season represented the strongest single-seater showing of his junior career.
He also competed at the 24 Hours of Le Mans twice during this period: in 1982, driving a Porsche 934/5 with Claude Haldi and Rodrigo Terran, the car retired after 141 laps; in 1983, driving a Lancia LC1 with Thierry Perrier and Bernard Salam, the car was classified non-classified after 133 laps.
Ferté had been expected to take the second Ligier seat for 1984, but it was Hesnault who was named as teammate to Andrea de Cesaris in the Ligier JS23-Renault V6 turbo. The choice came as a surprise given that his F3 résumé was considered modest for a factory seat. Out-qualified by de Cesaris on almost every occasion, Hesnault failed to score a championship point across the season. His best result was seventh place at the Dutch Grand Prix at Zandvoort; he also posted eighth places at the German and Austrian rounds. He retired from the Brazilian, Belgian, San Marino, Monaco, Canadian, Detroit, Dallas, Italian, and Portuguese Grands Prix.
For 1985, Hesnault was selected to partner Nelson Piquet at Brabham in the BT54-BMW turbo, a significant step up in quality compared to the Ligier. However, his three race starts at Brazil, Portugal, and San Marino were uncompetitive, and he failed to qualify at Monaco. Worse followed at a test session at Circuit Paul Ricard, where Hesnault suffered a major crash at the section known as the "S" de la Verrerie. The Brabham BT54-BMW was completely destroyed; Hesnault was trapped in the car while marshals worked to extricate him. Although uninjured physically, the accident shook him, and he and Brabham parted by mutual consent. Marc Surer replaced him at the team.
Hesnault made one final Grand Prix appearance at the 1985 German Grand Prix at Hockenheim, driving a third Renault RE60 entered by Équipe Renault Elf alongside the regular two-car entry. The car was fitted with a prototype onboard camera, making the 1985 German Grand Prix the first occasion the technology was used in a Formula One World Championship race. As a third entry from the same constructor, the car was not eligible to score championship points. Hesnault retired on lap 8 with a clutch failure. The 1985 German Grand Prix is also the last Formula One race in which three cars from the same team were entered; subsequent regulations prohibited the practice.
After this appearance, Hesnault retired from motor racing entirely. He subsequently settled in Switzerland.
Hesnault's Formula One career covered three teams — Ligier, Brabham, Renault — across two seasons, totalling 21 entries and 19 race starts. He scored zero championship points. Outside of Formula One, he accumulated six wins and 23 podiums across his junior and F3 career. His lasting contribution to the sport is the technological footnote of carrying the first in-race onboard camera in a World Championship Grand Prix.