Costa began his motorsport career in karting in 2002 at the age of ten, racing in various championships in Brazil. Over seven years he accumulated fifteen championships and finished runner-up to Leonardo Cordeiro in the national Petrobras Karting Championship in 2008. After receiving a scholarship from the Skip Barber Racing School in the United States, Costa made his formula racing debut in 2009 with a one-off appearance in the MSA Formula Ford Championship in the United Kingdom.
In 2010 Costa contested his first full season of car racing in Felipe Massa's Formula Future Fiat series; his championship victory earned him admission into the Ferrari Driver Academy, making him the first South American to become an academy member.
For 2011 Costa moved to the European single-seater ladder, competing in the European and Italian Formula Abarth series for Cram Competition, finishing ninth and fifth respectively. He was dropped from Ferrari's roster at year end. He remained in Formula Abarth for 2012, moving to Vincenzo Sospiri's Euronova Racing team operated by Fortec Motorsport, and won both the European and Italian championship titles over Luca Ghiotto. Costa also participated in GP3 Series post-season tests twice, driving for Carlin in 2011 and Marussia Manor Racing in 2012, but did not secure a race seat.
Costa moved to the United States' Road to Indy in 2013, competing part-time in the Pro Mazda Championship with Team Pelfrey, claiming four podiums. For 2014 he joined Dallas-based M1 Racing on a round-by-round basis before returning to Team Pelfrey for the remainder of the season; a pair of pole positions and a race win at Mid-Ohio, plus three second-place finishes, left Costa fifth in the final standings.
Costa made a surprise debut in the F4 Japanese Championship in 2015, rejoining Vincenzo Sospiri Racing alongside Takuro Shinohara as a junior driver for Lamborghini Squadra Corse. Despite a fourth-place finish in the opening round at Okayama, he was replaced by Gustavo Myasava before round five at Sugo. He subsequently tested a Lamborghini Huracán Super Trofeo alongside Shinohara at Adria International Raceway in October 2015.
Between 2016 and 2018 Costa raced for Vincenzo Sospiri Racing and Lamborghini in various GT series, partnering Yūki Nemoto and Ling Kang in the Italian GT Championship, Lamborghini Super Trofeo, and the International GT Open. He claimed the GTCup class championship in Italian GT in 2016 and scored a second-place finish in a one-off appearance at the 2017 Blancpain GT Series Asia at Suzuka alongside Sandy Stuvik.
Costa returned to Japan for the 2019 Super Taikyū endurance series, driving a B-Max-operated Nissan GT-R GT3 for Tairoku Racing; he finished on the podium at the Fuji 24 Hours with team owners Tairoku Yamaguchi and Shinichi Takagi, Satoshi Motoyama, and Harrison Newey before the team's withdrawal. He spent 2020 as a driver coach in Super Taikyū.
Costa returned to Brazil in 2021, competing in Porsche 911-based one-make championships over three seasons. After two part-time seasons in the Porsche Endurance Challenge Brasil, he moved to the Porsche Carrera Cup Brasil for 2023. He won the series' first round at Interlagos from pole position, becoming the first debut race winner since 2018, and went on to win six further races. Costa claimed the championship title by 28 points over four-time series champion Miguel Paludo.
In February 2024 Costa was announced as returning to international racing, piloting a McLaren 720S GT3 Evo for United Autosports in the FIA World Endurance Championship alongside Grégoire Saucy and James Cottingham.
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