Red Bull's purchase of Minardi at the end of 2005 ended twenty years of the Italian team's independent existence. The new owners retained Minardi's technical staff and factory at Faenza but not the former owner Paul Stoddart's other assets in Ledbury. The team adopted the Italian translation of the Red Bull brand name — Toro Rosso, meaning "Red Bull" — and inherited an arrangement allowing them to run V10 engines under an FIA equivalency formula rather than the V8 units mandated for all other competitors.
This agreement had been designed to benefit the financially limited Minardi rather than a company of Red Bull's scale, and it remained contentious throughout the season. The V10 dispensation allowed Toro Rosso to run an almost identical version of the Red Bull RB1 from 2005 paired with Cosworth engines, though these ran without factory support and under power restrictions imposed by the equivalency formula. The STR1 was also the first Toro Rosso car to feature the mandatory seven-speed gearbox configuration and the first to use Michelin tyres since the Minardi PS02 in 2002.
The V10's potential torque advantage was largely negated by traction control systems, and the use of a year-old chassis limited development options. Despite concerns that the V10 arrangement might give Toro Rosso an unfair advantage, those fears proved unfounded in practice. The team was generally more competitive than Midland, Spyker, and Super Aguri, but remained clearly separated from the midfield above them.
Liuzzi scored the team's only point of the season at the United States Grand Prix in Indianapolis. He was the faster of the two drivers for most of the year, though Speed improved steadily as the season progressed. Both drivers were involved in incidents and errors consistent with limited Formula One experience, but drew praise from part-team owner Gerhard Berger. The team finished ninth in the Constructors' Championship with one point.
Compared to the Red Bull Racing livery used by the sister team, the STR1 wore a darker shade of blue with a golden-tipped nosecone. A striking feature was a raging bull painting on the engine cover, designed by Austrian sculptor Jos Pirkner and hand-sprayed by Knud Tiroch before each Grand Prix. Sponsorship logos were minimal, limited to Red Bull, Michelin, and Cosworth.
After its Formula One career concluded, the STR1 was transferred to the BOSS GP Formula One series. As of 2025, the car was actively raced by Austrian driver Ingo Gerstl for Team Top Speed. Gerstl and the STR1 won the BOSS GP Open Class title in 2016, 2017, 2018, 2019, 2020, and 2022, making the STR1 one of the most successful vehicles in post-retirement Formula One competition.