Galli came from a wealthy textile family in Prato. He began club racing in the early 1960s with the Scuderia Santo Stefano of Prato, competing in hillclimbs and touring car events in cars including a Fiat-Abarth 700, an Alfa Romeo Giulietta Sprint Zagato, a Steyr-Puch 500D, and a Mini Cooper S. His ability caught the attention of Carlo Chiti, the engineer who founded Autodelta — Alfa Romeo's factory competition arm — and Galli was hired to join the works touring car team. The Autodelta roster at the time included Andrea de Adamich, and Galli's most important early partnership was with Ignazio Giunti, who became both his regular co-driver and a close personal friend.
He made his debut in international sports car racing in 1962 at the Coppa Maifredi at Circuito del Garda in a Fiat-Abarth 700. In 1966 he achieved his first WSC podium at Mugello, sharing the Autodelta Alfa Romeo Giulia Sprint GTA with Giunti, and also finished second at Zandvoort in the European Touring Car Championship.
In 1967 Galli traveled to Sebring for his North American debut in an Autodelta Alfa Romeo T33, though the car did not finish. He also won the ETCC race at Aspern and drove a Ferrari 275 GTB at the 1000 km of Monza — shared with a friend from Prato, finishing second in class.
The 1968 season was his most successful in endurance racing. He finished second at the Targa Florio with Giunti in the Alfa Romeo T33/2, behind the Porsche 907 of Vic Elford and Umberto Maglioli. At the Nürburgring 1000 km they finished fifth. At the 24 Hours of Le Mans, Galli and Giunti brought their Alfa Romeo T33/2 home fourth overall, winning the P2.0 class. That same season Galli won the outright victory at the Circuito di Mugello — a non-championship race on the old 66.2-kilometre road course — sharing with Vaccarella and Lucien Bianchi. On 19 July 1970, driving a Lola T210-Ford for Alain de Cadenet's Ecurie Evergreen, Galli set the absolute lap record on the Mugello road course — 29 minutes 36.8 seconds, averaging 134.128 km/h. The record was never beaten because the road course was abandoned in 1971; a plaque commemorating it was placed along the route in 2015.
In 1969 Galli joined the Tecno Racing Team alongside François Cevert in the European Formula 2 Championship and finished seventh in the standings. For Le Mans that year he drove for Equipe Matra Elf with Robin Widdows, finishing seventh. In 1970 his Alfa Romeo T33/3 partnership with Stommelen included a role in the filming of Steve McQueen's Le Mans movie.
Galli's Formula One debut came at the 1970 Italian Grand Prix at Monza, where he drove a McLaren M7D with Alfa Romeo power. He failed to qualify. He made his first championship start at the 1971 Dutch Grand Prix with STP March, driving the March 711. His 1971 season produced scattered finishes — 11th in Britain, 12th in Germany — with a career-best qualifying position of 15th at the Austrian Grand Prix.
For 1972 Galli joined the Martini Racing-backed Tecno team, sharing the troublesome flat-12 PA123 with Derek Bell. The car retired from most rounds. He finished third in the non-championship Grand Prix della Repubblica Italiana at Vallelunga. Two weeks later, Scuderia Ferrari invited him to deputise for Clay Regazzoni — injured — at the French Grand Prix at the Circuit de Charade near Clermont-Ferrand. Qualifying 21st of 24 starters in the Ferrari 312B2, Galli finished 13th. It was his sole Ferrari drive.
For 1973 Galli moved to Frank Williams Racing Cars, driving the Iso-Marlboro with Howden Ganley as a teammate. He contested five races; his best result was ninth at the 1973 Brazilian Grand Prix at Interlagos. After retiring from the Monaco Grand Prix with a halfshaft failure, he announced his retirement from Formula One. Across 20 championship entries and 17 starts he scored zero points.
Galli made a brief return in 1974 with an Abarth-Osella PA2 at the Monza and Imola rounds of the World Championship for Makes, before retiring entirely at age 33. His final race ended when Arturo Merzario — driving an Alfa Romeo 33TT12, a marque to which Galli had given most of his career — put him out at Imola. After retirement he entered the clothing business, becoming the Italian distributor for Fruit of the Loom. His company was a minor Williams Racing sponsor, and in 1983 he was instrumental in bringing the Benetton Group into Formula One.
In 2014 Galli returned to Le Mans for the Le Mans Classic. He died on 12 October 2019 in Prato, aged 79, ten days after his birthday, following a long illness.