Courage was born into a prominent English family: his father was chairman of the Courage brewing dynasty, and Piers was educated at Eton College. He began racing in his own Lotus 7, and after a brief introduction to the European Formula 3 circuit in 1964 with a Lotus 22 alongside Jonathan Williams, he pursued a full season in 1965. Driving a 1.0-litre Formula 3 Brabham for Charles Lucas, he formed a pivotal early alliance with Frank Williams, then one of Lucas's other drivers. Four high-profile wins in 1965 prompted Colin Chapman to offer Courage a Lotus 41 for the 1966 Formula 3 season, though the car proved inferior to the dominant Brabhams. He was nonetheless given a step up to Formula 2 at the 1966 German Grand Prix.
BRM signed Courage for its works Formula One team for 1967 alongside Chris Irwin. His aggressive driving style led to several accidents, and BRM released him after the 1967 Monaco Grand Prix. He spent the remainder of the season in John Coombs's Formula 2 McLaren M4A, purchasing the car at the season's end. A strong run in the McLaren during the winter Tasman Series, including a race victory at the final round, earned him a seat in Tim Parnell's works-supported Reg Parnell Racing BRM team for 1968, a season that included points-scoring finishes in France and Italy.
When Frank Williams Racing Cars stepped up to Formula One for 1969, Courage was the natural choice of driver. The car was a Brabham BT26, painted in the dark blue that became associated with the team, and in Courage's hands it proved competitive beyond what its private-team resources should have permitted. At the Monaco Grand Prix he finished second, and he repeated the result at the United States Grand Prix at Watkins Glen. Perhaps his most impressive individual performance was the 1969 Italian Grand Prix at Monza, where he held station with the leading group for the majority of the race in an older car before fuel starvation slowed him near the end. He finished fifth. A second fifth place at the British Grand Prix completed a season that yielded 16 championship points and eighth place in the standings.
For 1970, Williams switched from Brabham to a newly designed De Tomaso chassis following an arrangement with Alejandro de Tomaso. The car was heavy and unreliable, and a promising 1969 trajectory stalled sharply. At the Dutch Grand Prix at Zandvoort, Courage qualified ninth. Running in the middle of the field, the De Tomaso's front suspension or steering failed on the bump at Tunnel Oost, causing the car to go straight on through a high-speed bend. The machine rode up an embankment and disintegrated on impact, the engine separating from the monocoque as the wreckage burst into flames. To save weight, magnesium had been used in the car's chassis and suspension components; the material burned intensely, setting nearby trees and bushes alight. Courage was fatally injured when a front wheel broke free and struck him in the head. He died at the circuit on 21 June 1970, aged 28.
He was survived by his wife, Lady Sarah Marguerite Curzon, and two sons. He was buried at St Mary the Virgin churchyard in Shenfield, Essex.
The same corner where Courage was killed — the Tunnel Oost bump at Zandvoort — claimed Roger Williamson's life just three years later in 1973. Courage's death, along with a series of other fatalities in that period, contributed to growing pressure for improved safety standards in Formula One. His 1969 season with Frank Williams remained a demonstration of what could be achieved with a private team and a skilled driver operating in genuine partnership, a model Williams would build upon long after Courage's death.