The STR3 was designed by Adrian Newey, the Red Bull Racing designer whose cars formed the technical basis for Toro Rosso's preceding machinery. The car was first tested at a circuit in Italy on 2 April 2008 by Red Bull junior driver Brendon Hartley, before being publicly launched on 16 April 2008 at the Circuit de Catalunya, where Sébastien Bourdais gave it its first public appearance. The STR3 visually and technically resembled the Renault-powered Red Bull RB4, continuing the close technical relationship between the two Red Bull-owned teams.
Toro Rosso had originally targeted introduction of the STR3 as early as the fourth or fifth round of the season. However, the debut was pushed back to the Monaco Grand Prix — round six — because chief engineer Laurent Mekies, who would later become team principal of Toro Rosso's successor identity Racing Bulls and then of Red Bull Racing, determined that the team did not have sufficient parts to run the programme professionally in Turkey. The STR2B consequently remained in service through the opening five races.
The STR3 made its race debut at Monaco driven by both Sébastien Bourdais and Sebastian Vettel. From that point the car showed improvement over its predecessor, gradually finding pace through the mid-season and beyond.
The defining moment of the STR3's existence came at the 2008 Italian Grand Prix at Monza. In changeable and demanding conditions, Sebastian Vettel delivered a masterful performance to claim pole position and convert it into a race victory — the first of his Formula One career. It was also the only race victory in the history of the Toro Rosso team. The successor team identity, Scuderia AlphaTauri, did not achieve a second win until Pierre Gasly's victory at the 2020 Italian Grand Prix at the same circuit, twelve years later.
Toro Rosso finished sixth in the Constructors' Championship in 2008 with 39 points across the whole season — the team's best championship result since 1991 under the Minardi name. Of those 39 points, 37 were scored with the STR3. Vettel ended the year with 35 points and Bourdais with 4.
Vettel, who had a habit of naming his racing cars, gave the STR3 the name Julie.
A chassis from the STR3 programme — specifically the car driven by Vettel — made a return to competitive action in the 2022 BOSS GP Series. Thomas Jackermeier, the CEO of German sim racing hardware company Fanatec, drove the car in the Formula One class at the Red Bull Ring, scoring two second-place finishes.
The STR3 cemented Sebastian Vettel's reputation as an exceptional talent and accelerated his move to Red Bull Racing for the 2009 season, where he would go on to win four consecutive World Drivers' Championships. For Toro Rosso, the Monza victory stands as the high-water mark of the team's entire existence, a result achieved by a junior squad against the established order of Formula One in one of the sport's most storied venues.