Lavaggi was born into a noble Sicilian family whose lineage traces to Genoa in the 1400s. He studied mechanical engineering at Milan Polytechnic, a qualification that would shape the technical dimension of his racing career. His first involvement in motorsport administration came in 1992, when he served as an official test driver for the March Formula One team, drawing on his engineering background. His racing education began through Henry Morrogh's racing school in 1984, where Morrogh judged him the best student he had ever taught.
Lavaggi started competing in 1984 in the Italian Formula Panda championship driving for constructor Ermolli, finishing second overall and winning more races than any other driver in the series. Without sufficient funding for a full Formula Three season, he contested only a handful of races in Italian Formula Three before switching to Group C sportscars to gain international experience, joining the Porsche Kremer Team.
In Group C and its successors, Lavaggi proved consistently competitive. He became the 1993 Interserie champion, winning six of the twelve races in the season and adding four further podium finishes. He also won the 1995 Daytona 24 Hours as part of a four-driver team, driving nine hours himself. In the FIA Sportscar Championship he scored two wins, including the 1000 km of Monza where he drove five of the race's six hours, along with five more podiums and two pole positions.
Lavaggi's Formula One debut came at the 1995 German Grand Prix, driving for the Pacific team. He had tested the Pacific PR02 twice previously, at Silverstone and Snetterton. At 35 he was the oldest driver in the field — a fact that, combined with the unconventional path to F1, attracted some scepticism from the paddock press. The Pacific PR02 was one of the least competitive cars in the field and retired from mechanical failures in all four of his 1995 appearances, each time from gearbox problems.
For the second half of the 1996 season, Lavaggi joined the Minardi team for six races. His best result was a tenth-place finish at the Hungaroring, which represented the second-best result of the year for the Minardi team across their entire season.
Away from race weekends, Lavaggi competed at the 1996 Bologna Motorshow event, racing a Minardi against cars from Benetton, Ligier, and his Minardi teammate Tarso Marques. He finished second, losing the final to Giancarlo Fisichella by a very narrow margin.
Following his Formula One years, Lavaggi returned to endurance racing. He won a race at Monza in the 2001 FIA Sportscar Championship and continued in the Le Mans Endurance Series in 2004 and 2005.
In 2006, Lavaggi founded his own outfit, Scuderia Lavaggi, and crossed from driver to constructor by designing and building the Lavaggi LS1, a Le Mans Prototype. He raced the car in the Le Mans Series through 2009, completing a journey from engineering student to racing driver to manufacturer within a single career.
Lavaggi was nicknamed "Johnny Carwash" by fellow paddock inhabitants — an approximate English translation of his Italian name — a nickname later brought to broader public attention by US television host David Letterman. The path from late starter to F1 entrant to independent constructor, achieved without personal wealth, made his career one of the more idiosyncratic in the sport's modern era.