Günther began karting in 2007 but transitioned to open-wheel racing by 2011, competing in the Formula BMW Talent Cup. After a year away from racing in 2012, he entered the ADAC Formel Masters in 2013, finishing second in the standings and claiming his first victories at the Lausitzring. He was runner-up again in 2014 before stepping up to the FIA Formula 3 European Championship in 2015.
In Formula 3, Günther raced for Mücke Motorsport in his debut year before switching to Prema Powerteam. With Prema in 2016, he took four victories including wins at Paul Ricard and Monza, ultimately finishing as championship runner-up to Lance Stroll by 68 points. He continued with Prema in 2017 alongside Callum Ilott, Guanyu Zhou, and Mick Schumacher, winning five races including a double at Pau and another at the Norisring to take third in the championship, five points behind runner-up Joel Eriksson. Champion that year was Lando Norris.
For 2018, Günther graduated to Formula 2 with BWT Arden, winning the Silverstone sprint race from reverse-grid pole position. He finished 14th in the championship before departing mid-season after signing with Dragon Racing in Formula E.
Günther served as Dragon's test and reserve driver during the 2017–18 season before being promoted to a race seat for 2018–19 following Jérôme d'Ambrosio's departure. After three races in which he was temporarily dropped and replaced by Felipe Nasr, Günther returned and scored fifth place at the Paris ePrix and again at the Swiss ePrix, finishing 17th in the standings with 20 points in his debut season.
Günther moved to BMW i Andretti for 2019–20, replacing António Félix da Costa. At the second race in Diriyah, he provisionally finished second but was demoted to 11th for a safety car infringement. At the following race in Santiago, he overtook da Costa on the final lap to take his maiden Formula E victory, becoming the youngest ever winner in the series at that point. He added a second win in the third Berlin race during the COVID-19-curtailed season finale, finishing ninth in the championship with 69 points, all three of his scoring finishes being podiums.
In 2020–21, Günther was retained alongside Jake Dennis. Despite being disqualified from one race in Rome after a qualifying crash, he won the New York City ePrix following a late collision between Jean-Éric Vergne and Nick Cassidy. He finished 16th in the championship with 66 points, only 33 behind champion Vergne.
Günther joined Nissan e.dams for the 2021–22 season, partnering champion Sébastien Buemi. The season was difficult; outside of a points finish in Mexico City and a top-ten in London, the team lacked the pace to compete consistently. Günther ended the season 19th with just six points.
Günther switched to Maserati MSG Racing for 2022–23, partnering Edoardo Mortara. After a difficult start to the campaign, a breakthrough came at the Jakarta ePrix where Günther claimed his maiden Formula E pole position for both races and dominated Sunday's race for Maserati's first single-seater win since the 1957 German Grand Prix. He added podiums in Rome and Portland and finished seventh in the championship with 101 points, his best Formula E season to date.
In 2023–24, partnering new teammate Jehan Daruvala, Günther won the Tokyo ePrix from the front row and added a podium at Misano. He finished eighth in the standings with 81 points before departing Maserati at the season's end.
Günther joined DS Penske for the 2024–25 season alongside Jean-Éric Vergne. He won the Jeddah pit-boost race with a last-lap overtake on Oliver Rowland and later claimed the first race victory at the Shanghai ePrix, managing his second attack mode activation to pass Rowland in the closing stages. He finished tenth in the championship. For 2025–26, he was joined by new teammate Taylor Barnard.
Günther holds dual German and Austrian nationality and resides in Monaco. He speaks German, English, French, and Spanish.