By 2004 Jaguar Racing had stabilised its management structure under Tony Purnell and David Pitchforth following years of upheaval. The R4 of 2003 had brought a sixth-place Constructors' Championship finish and demonstrated that the team was capable of competent mid-field racing. The R5 was developed as a continuation of that work, though the programme was overshadowed from almost the start of the year by speculation about the team's future. Ford had been reviewing the cost and strategic value of its Formula One involvement, and the writing was on the wall long before the season ended.
The R5 was a genuine step forward from its predecessor, featuring a revised chassis and aerodynamic package. An updated R5B specification was also prepared and used selectively across two races: Christian Klien gave the B-spec its competitive debut in China, and Webber drove it in Brazil. The R5B had originally been transported to the Italian Grand Prix as a possible option but was not raced there. Webber also ran the R5B during Friday practice in Japan before reverting to the standard specification for the race due to limited running time.
Engine supply continued through Cosworth as a Jaguar-badged works customer arrangement, providing a reliable if not class-leading power unit throughout the year.
Webber proved the more productive of the two drivers, recording four points finishes during the year including a sixth place at the Hockenheimring. Klien, in his first Formula One season, showed encouraging consistency, retiring on only four occasions and achieving his own sixth-place result at the Circuit de Spa-Francorchamps in Belgium. The team finished seventh in the Constructors' Championship with ten points, a modest outcome that nonetheless reflected steady if unspectacular competitiveness.
At the Monaco Grand Prix Jaguar adopted a promotional livery to tie the car to the release of the film Ocean's Twelve. The airbox and nose featured the film's logo, and a Steinmetz diamond worth $300,000 was affixed to the nose of Klien's car. Klien crashed at the Loews hairpin on the opening lap, and when Jaguar personnel reached the wreck, the diamond had disappeared, generating a wave of international media attention that briefly made the team front-page news far beyond the motorsport press. At the Brazilian Grand Prix, two Jaguar engineers adopted an inflatable donkey mascot from a promotional campaign for the film Shrek; the item was signed by Bernie Ecclestone, Max Mosley, and every Formula One driver except Michael Schumacher, before being auctioned for charity.
Ford sold the Jaguar Racing operation to Red Bull in November 2004. The acquisition came with very little lead time before the first pre-season tests of 2005, and the new owners repainted the R5 in a light blue and grey livery resembling an energy drink can to tide them over until a bespoke car could be built. Red Bull retained Christian Klien for their first season while Webber moved to Williams, though he would later return to the team, by then renamed Red Bull Racing, and go on to become a race winner and championship contender.
The R5 chassis later appeared in historic racing under the BOSS GP banner. At the 2011 Hockenheim Historic, a Jaguar R5 driven by Klaas Zwart finished second in the opening race before retiring from the second. One of the surviving cars was badly damaged during a BOSS GP round at the TT Circuit Assen in 2017 but was subsequently repaired and returned to competition in 2019.