Kinnunen began on motorcycles before switching to four wheels in the early 1960s, competing in rallying, autocross, and ice racing on Finnish roads. He finished runner-up in the Finnish Rally Championship, matching the points total of winner Simo Lampinen, and raced in the Finnish Formula Three Championship with a Brabham in 1967 and a Titan in 1968. His talent on circuits became apparent when he defeated future Formula One star Ronnie Peterson at Ahvenisto Race Circuit.
The decisive step came in 1969 when Kinnunen won the Nordic Challenge Cup — predecessor of the Interserie — at Keimola Motor Stadium, beating Jochen Rindt in close combat. That performance earned him an invitation to test a Porsche 917 at Österreichring, and his times were sufficiently impressive that he was hired to partner Pedro Rodriguez in the World Sportscar Championship. The pair opened the 1970 season by winning the 24 Hours of Daytona, and Porsche took the overall manufacturers' championship. Kinnunen's relationship with the programme was complicated, however: the car had been built around Rodriguez's preferences, and once Kinnunen began outpacing his teammate on his own set-ups, he was prevented from making further adjustments. His clearest individual moment came at the Targa Florio, where Rodriguez was ill and Kinnunen drove solo on the final lap, setting a time of 33 minutes 36 seconds that broke the previous circuit record by ninety seconds — a mark that stands as the fastest ever recorded on the Sicilian road course.
Moving to the Finnish AAW Racing Team for the Interserie, Kinnunen won the championship in his debut year despite withdrawing from the Norisring round after watching Rodriguez die in a crash at the barriers. He went on to win the title three consecutive times, in 1971, 1972, and 1973, accumulating 18 heat wins and 11 outright victories across the series. In 1973 he also contested the 1000 Lakes World Rally Championship round in a Porsche Carrera, finishing third behind Timo Mäkinen and Markku Alén.
Kinnunen had come close to a Formula One seat for 1971 after Jochen Rindt recommended him to Team Lotus, but Rindt's death at Monza interrupted those negotiations, and terms could not be agreed. In 1974, John Surtees offered him a Surtees TS16 on lease, run under the familiar AAW Racing name with a Cosworth DFV engine.
The campaign was beset by mechanical and financial misfortune from the start. At the Belgian Grand Prix at Nivelles, the TS16 arrived with a damaged monocoque, faulty rear suspension, and a weight excess of 80 kilograms, and its gearbox failed after just three laps of practice. A further gearbox failure in qualifying left Kinnunen without a recorded time — the only driver not to qualify. The team skipped Monaco and entered the Swedish Grand Prix, where an exception was made to allow Kinnunen to start from 25th place given his standing in Swedish motorsport circles. Running light on fuel for only ten laps, he overtook five cars before a stripped spark-plug thread ended his race. The Dutch Grand Prix entry was refused outright, and the team failed to qualify in France, Great Britain, Italy, and Austria. Funding dried up and the programme was abandoned.
Kinnunen is noted as the last driver to compete in Formula One wearing an open-face helmet.
After his Formula One stint, Kinnunen returned to the Interserie for its final Hockenheim event, winning both heats. He was then engaged by Martini Racing for the 1975 World Sportscar Championship in a Porsche 908 Turbo, taking third at the Nürburgring alongside Herbert Müller. The following season he raced a Porsche 934 Turbo with Egon Evertz, achieving a second-place finish at Watkins Glen before that team also withdrew for financial reasons. He retired from circuit racing in 1977, though he continued to contest Finnish rally events, winning the 1979 Arctic Rally. He remained active in motorsport administration, living in Turku until his death in July 2017.
Steve McQueen personally invited Kinnunen to drive in the 1971 film Le Mans, but Kinnunen's Porsche contract prevented his participation; he was replaced by David Piper, who was seriously injured during filming.