Kvyat began competitive karting aged ten and moved to Italy in 2007 to pursue a professional racing career. His karting results caught the attention of Red Bull, and he joined the Red Bull Junior Team in 2010. He progressed to formula cars in Formula BMW, then Formula Renault 2.0, where in 2012 he won the Alps Series championship and finished runner-up to Stoffel Vandoorne in the Eurocup despite a controversial tyre-choice call by his Koiranen GP team in the Barcelona finale that cost him the title.
For the 2013 season Kvyat joined MW Arden in the GP3 Series alongside a guest programme in the European Formula 3 championship with Carlin, where he was ineligible for points as a late entry. In GP3 he quickly found his footing, claiming his first series victory at Spa in August and reducing the gap to the championship leaders. He headed into the Abu Dhabi season finale as a genuine title contender.
Kvyat's performances at the final two rounds at Monza and Abu Dhabi were defining: he took pole position, led every lap, set the fastest lap, and won both feature races, accumulating maximum points at each event. He won the 2013 GP3 Series championship in his debut season. The title clinched his promotion to Formula One, and Helmut Marko offered him an F1 contract following a dominant win at Zandvoort in F3, where the organisers were caught without a recording of the Russian national anthem and substituted Mikhail Glinka's "The Patriotic Song" at the podium ceremony.
Kvyat joined Scuderia Toro Rosso for 2014, making his debut at the Australian Grand Prix aged nineteen. He finished ninth, breaking Sebastian Vettel's record as the youngest points-scorer in Formula One history at the time. He was promoted to Red Bull Racing for 2015, where he scored his maiden podium with second place at the Hungarian Grand Prix, becoming at 21 years and 91 days the second-youngest driver to stand on an F1 podium behind Vettel. He outscored teammate Daniel Ricciardo by three points to finish seventh in the 2015 championship.
The 2016 season turned dramatically at his home Russian Grand Prix, when Kvyat made contact with Sebastian Vettel twice in a single incident. Red Bull demoted him back to Toro Rosso in place of Max Verstappen from the very next race in Spain. After a troubled 2017 campaign at Toro Rosso, he was released from the Red Bull programme entirely.
Following a year as a Ferrari development driver in 2018, Kvyat returned to the renamed Toro Rosso for 2019 and delivered what many considered his finest F1 season. At the rain-affected German Grand Prix he pitted for dry-weather tyres ahead of the rest of the field and overtook Lance Stroll for a shock third-place finish — the team's first podium since their victory at the 2008 Italian Grand Prix. He was not retained for 2021, with Yuki Tsunoda taking his seat.
After serving as Alpine's reserve driver in 2021, Kvyat joined Prema Racing in the FIA World Endurance Championship LMP2 class for 2023, racing under an Italian licence. He scored a podium on debut at the 1000 Miles of Sebring. In 2024 he moved to Lamborghini in the Hypercar class, and in 2026 made his Super GT debut in the GT300 class for the JLOC team at the wheel of a Lamborghini Huracán GT3 Evo 2.
Kvyat's career is defined by extremes: brilliant pace and racecraft that earned him three F1 podiums, contrasted with the turbulent 2016 Russian Grand Prix episode that led to his famous mid-season demotion. He remains the highest-achieving Russian driver in Formula One history, and his 2013 GP3 title stands as one of the most dominant rookie campaigns in the series.