Oliver began motorsport in 1961 driving a Mini in British club racing, progressing through Formula Three and GT events in the mid-1960s. His natural speed was frequently undermined by mechanical failures in junior formulae. For 1967, he was drafted into the Team Lotus Formula Two team and made his Formula One debut in the F2 class at the German Grand Prix, coming fifth overall and winning the F2 category.
In 1968, following the death of Jim Clark, Colin Chapman called Oliver into the Lotus Formula One seat. The season was difficult but contained a memorable moment: Oliver led the British Grand Prix until engine failure ended his race. He finished third at the season-closing Mexican Grand Prix, his best Formula One result. With Jochen Rindt joining Lotus for 1969, Oliver moved to BRM, where two disappointing seasons followed. Though he was capable of running competitively — he led much of the 1970 Race of Champions, held strong positions at several Grands Prix, and scored points at Mexico in 1969 and Austria in 1970 — mechanical failures were constant, and BRM released him.
In 1973 Oliver became team leader for the new Shadow Formula One team. The Shadow DN1 was a difficult car and results were limited, though Oliver drove a strong race in Canada that year — many contemporary observers believed he actually won, but confusion surrounding the timing of a rain shower and a poorly deployed pace car saw him officially classified third. He returned briefly to Formula One for Shadow in subsequent seasons and contested the 1977 Race of Champions, finishing fifth, before his role shifted to the team management side.
Oliver competed in 52 Formula One World Championship Grands Prix in total, achieving two podiums.
Oliver's most celebrated victories came in endurance racing with John Wyer's Gulf-sponsored operation. In 1969, paired with Jacky Ickx in a Ford GT40, he shared one of the most dramatic finishes in Le Mans history: the closing hours produced a lap-by-lap duel with the Porsche 908 of Hans Herrmann and Gerard Larrousse, Ickx eventually prevailing by fewer than 120 yards after having started last. The same year, Oliver and Ickx won the 12 Hours of Sebring together.
At the 1971 24 Hours of Daytona, Oliver won alongside Pedro Rodriguez, completing his endurance Triple Crown. He also co-drove with Rodriguez to victory at the 1000 km Monza that year.
In Can-Am, Oliver initially drove for the Shadow team from 1969 onward and became increasingly central to both its racing and administrative operations. He took the Canadian-American Challenge Cup title for Shadow in 1974, his most significant single-seater championship.
Oliver also competed in eight NASCAR Cup Series races between 1971 and 1972, attempting the 1972 Daytona 500 but failing to qualify.
At the end of 1977, Oliver departed Shadow along with financier Franco Ambrosio, designers Tony Southgate and Alan Rees, engineer Dave Wass, and driver Riccardo Patrese to found the Arrows Grand Prix team. Oliver served as team principal through the team's entire existence. Arrows became notable for competing in a record 382 Grands Prix without winning any of them. Oliver sold much of his stake to the Footwork Corporation in 1990 and remained as director; after Footwork withdrew at the end of 1993, he regained control before selling the majority of his shares to Tom Walkinshaw's TWR group in 1996. He remained on the board until 1999.
Oliver occupies an unusual position in motorsport history: a driver of genuine Formula One and endurance calibre who became one of the longest-serving team principals in the sport's history with Arrows. His 1969 Le Mans victory alongside Ickx is regarded as one of the great endurance races, notable as much for the drama of the finale as for the improbable nature of the ageing Ford GT40's defeat of newer and faster machinery. His Can-Am title and the endurance Triple Crown confirm the breadth of a career that spanned driving, team management, and team ownership across four decades of top-level competition.