The STR13 came about as a direct consequence of the collapse of the McLaren-Honda partnership at the end of 2017. When McLaren terminated the deal, Renault confirmed it lacked the capacity to supply both McLaren and its existing customer teams simultaneously. Negotiations between all parties resulted in Toro Rosso signing a factory works agreement with Honda that was projected to run through 2021. Pierre Gasly and Brendon Hartley were retained as drivers, both having contested selected events in 2017.
The car's most consequential design decision was the Honda power unit. During pre-season testing in Barcelona, Honda demonstrated a dramatic reliability improvement over its troubled McLaren era: Toro Rosso completed more laps in the first week of testing than Honda had managed across the entire two-week test with McLaren in 2017. Hartley and Gasly praised the unit as more powerful and easier to manage than the Renault engines they had used previously.
The first race in Australia exposed a known Honda weakness: an MGU-H failure sidelined Gasly after contact with a kerb. Honda clarified that both drivers were still running the previous 2017-specification MGU-H unit, as the new 2018 component had not been ready in time for the opener. Bahrain brought the updated MGU-H for both cars, though Gasly required an entirely new power unit after collateral damage from the Australian failure.
The Bahrain Grand Prix provided Toro Rosso's best result of the year when Gasly qualified sixth and converted that into a fourth-place finish — Honda's highest result since its 2015 return to Formula One. The positive performance established early optimism about the Honda partnership.
Honda introduced a Spec 2 power unit at the Canadian Grand Prix, concentrating development on the internal combustion engine, specifically combustion chamber design. Gasly reported it as immediately punchier, and lap-time comparisons at Circuit Gilles Villeneuve indicated a roughly half-second gain. Post-summer, however, correlation difficulties between aerodynamic updates and the power unit complicated the team's ability to unlock consistent pace across different circuit layouts.
Honda debuted the highly anticipated Spec 3 power unit during practice at the Russian Grand Prix. Drivers noted a clear power increase but reported prominent vibrations on upshift when the new unit was mated to the Toro Rosso gearbox. The team reverted to Spec 2 for the race weekend while engineers refined the integration, before making the competitive Spec 3 debut at the Japanese Grand Prix — Honda's home race. There, both STR13s advanced to Q3, qualifying sixth and seventh, one of Honda's strongest qualifying showings since returning to the sport.
Post-season estimates from the German publication Auto Motor und Sport, derived from GPS data at Monza, placed the Honda Spec 2 at approximately 715 hp without electrical assistance, ahead of Renault's equivalent Spec B but behind Renault's new Spec C variant used exclusively by Red Bull Racing. Red Bull principal Christian Horner, while announcing a two-year Honda engine deal for Red Bull from 2019, cited the STR13 campaign as evidence of Honda's development pace, claiming the two power units were within one percent of each other in race trim by season's end.
Toro Rosso finished ninth in the Constructors' Championship with 33 points, the team's lowest single-season points total since 2012. Despite the modest championship return, the STR13's role as a test bed for the Honda partnership was widely viewed as successful, laying technical groundwork that led directly to Red Bull's adoption of Honda power and Honda's eventual championship success with the senior team.
In 2020, Yuki Tsunoda conducted a test in the STR13 — dressed in AlphaTauri AT01 livery at Imola — as part of his evaluation ahead of joining what had by then been renamed Scuderia AlphaTauri.