Crivillé was born in Barcelona. He was so eager to begin racing that he falsified his age to start competing at 14 in 1985, a year below Spain's minimum licence age of 15. He won the Criterium Solo Moto, a national series for 75cc Honda streetbikes, that year. After excelling in the 80cc World Championship with Derbi — finishing second in 1988, 47 points behind champion Jorge Martínez — he claimed his first world title in the 1989 125cc season with JJ Cobas, winning five races to take the championship by 14 points over Herri Torrontegui Spaan.
After two modest seasons in the 250cc class, Crivillé moved to the 500cc category with the Pons Racing Honda team for 1992. In only his third 500cc race, at the Dutch TT in Assen, he made history by becoming the first Spaniard to win a 500cc Grand Prix, edging John Kocinski by 0.762 seconds. He finished eighth in the championship that year.
For 1994, Crivillé joined the Factory Honda team alongside Mick Doohan, becoming the first Spanish rider to race for Honda's factory squad. He finished sixth in the championship with three podiums. In subsequent seasons he became one of Doohan's most consistent challengers within the team, finishing fourth in 1995 and second in 1996 — 64 points behind Doohan but ahead of all other rivals. The 1996 season produced some of the most dramatic internal team battles, including Crivillé winning the Austrian GP after Doohan made a small error and the Czech Republic GP by just 0.002 seconds — a margin that stood as a race record.
In 1997 a severe crash during qualifying at the Dutch GP left him with torn artery damage, a badly injured left thumb, and required skin and bone grafts. He missed five rounds but returned for the Czech Republic race, finishing fourth, and closed the season with victory in Australia. In 1998, Crivillé won in Spain — dedicating the victory to his father, who had died that winter — and in France. He finished third in the championship.
Few anticipated that Crivillé could challenge for the 1999 title after Doohan had claimed five consecutive championships, but Doohan's career-ending crash during qualifying at the Spanish Grand Prix in Jerez fundamentally altered the season's dynamics. Crivillé seized the opportunity and won five races across the year, including four in succession — Spain, France, Italy, and his home race in Catalunya, where he overtook teammate Tadayuki Okada on the final lap to win by 0.061 seconds in front of his home fans. He also won in his 100th 500cc start at the British Grand Prix at Donington Park.
Going into the penultimate round in Rio de Janeiro, Crivillé held a 44-point lead over Okada. Still recovering from a broken wrist sustained at Phillip Island, he started eleventh and needed to finish tenth or higher. He crossed the line sixth, ahead of Okada, and was declared the 500cc World Champion — the first Spaniard and first Catalan to hold that title. He finished the season with 267 points, 47 ahead of Kenny Roberts Jr.
The 2000 NSR500 proved more difficult than its predecessor and Crivillé struggled for form throughout the season, although he did take his final career victory at the French Grand Prix at Le Mans. He finished ninth in the championship. His final season, 2001, was hampered by two crashes at the German Grand Prix that left him with a broken finger and facial injuries; he recovered to take a podium at the Czech Republic but retired from four rounds. He finished eighth in the championship.
In September 2001, Crivillé publicly hinted at uncertainty over his future. On 12 November 2001 he confirmed he had parted ways with Repsol Honda after ten years. Negotiations with Aprilia and Yamaha's d'Antin team did not conclude with a deal, and in late 2001 he disclosed that a recurring medical condition — episodes of temporary loss of consciousness since 1999 — had forced a break from the sport. On 5 May 2002, at the Spanish Grand Prix, he held an emotional press conference and announced his permanent retirement, citing his illness.
Crivillé's 1999 title opened a new chapter for Spanish and Catalan motorsport, inspiring a generation of riders that would go on to dominate the sport's premier class across the following two decades. His combination of mechanical mastery and racecraft, developed through years as Mick Doohan's closest factory teammate, produced some of the tightest on-track battles in 500cc Grand Prix history.