Chaves began racing in 2007 in the Skip Barber series in the United States. His pro debut came in 2008 in Formula BMW Americas, and he also finished fourth at the Formula BMW Pacific round and fourteenth in the Formula BMW World Finals. In 2009 he raced a full Formula BMW Americas season for Eurointernational, winning the title with five victories and three pole positions.
Chaves then moved to Europe. In 2010 he competed in the Italian Formula Three Championship with Eurointernational, finishing tenth in the standings and taking Rookie of the Year honours. The following year he joined the Addax Team in the GP3 Series, finishing nineteenth in points with a best result of fourth at the Valencia Street Circuit.
In 2012 Chaves returned to the United States, signing with JDC MotorSports for the Star Mazda Championship. He finished second in points, taking wins in the final two races of the season at Mazda Raceway Laguna Seca and Road Atlanta.
He stepped up to Indy Lights in 2013 with Schmidt Peterson Motorsports, finishing runner-up with one win and ten podiums across the season. The year's Indianapolis Freedom 100 was particularly notable: Chaves was beaten by Peter Dempsey by just 0.0026 seconds in a four-wide photo finish, one of the closest in Indianapolis Motor Speedway history.
Chaves joined Belardi Auto Racing for the 2014 Indy Lights season and claimed the championship with four wins — including the Freedom 100 at Indianapolis — and eleven podiums from fourteen races. The title earned him a promotion to the IndyCar Series for 2015.
Chaves made his full IndyCar debut with Bryan Herta Autosport in 2015. His most prominent race of the season came at Pocono Raceway, where he led 31 laps after race leader Sage Karam crashed. That crash fatally injured fellow competitor Justin Wilson. Chaves resumed the lead when racing restarted with six laps remaining but was passed before his engine failed three laps from the end, denying him a podium. Despite missing the finish, he won the IndyCar Rookie of the Year award by 78 points over Stefano Coletti. A sponsor default meant Bryan Herta Autosport could not retain him for 2016, with Alexander Rossi taking his seat.
In 2016 Chaves drove part-time for Dale Coyne Racing, with a best result of twelfth at Detroit Race 1. He returned to Indianapolis in 2017 with Harding Racing for the Indianapolis 500 and additional oval rounds at Texas and Pocono.
Chaves was later appointed driver coach for USF Juniors, the entry-level series of the Road to Indy, ahead of its inaugural 2022 season.
Alongside his Indy Lights programme in 2014, Chaves joined the DeltaWing entry for the four endurance rounds of the United SportsCar Championship. He led fifteen laps in the Petit Le Mans, the 10-hour season finale at Road Atlanta, helping the team to a fourth-place finish — a season-best result for the DeltaWing programme.
From 2020 onward Chaves transitioned primarily to sports car competition. He joined Bryan Herta Autosport for a full season in the Michelin Pilot Challenge as co-driver with Ryan Norman, and has continued to compete in the WeatherTech SportsCar Championship and Michelin Pilot Challenge in subsequent seasons.
Chaves is one of a generation of drivers who bridged the Road to Indy ladder and the IndyCar Series in the mid-2010s. His 2014 Indy Lights title was comprehensive, and his 2015 IndyCar rookie season — interrupted by a sponsor default rather than any lack of performance — demonstrated competitive pace on both ovals and road courses. His subsequent pivot to endurance sports car racing has kept him active in North American professional motorsport.