Da Costa started karting in 2002 where he went on to win both the Portuguese Championship Cadet and the Portuguese K Open Cadet. He soon went on to take runner up in both the 2005 Portuguese Cup Junior and the 2006 ICA Junior WSK International Series. He then went on to win his last karting championship in the 2006 Junior Portuguese Championship before taking third in the 2006 ICA Junior Italian Open Masters and second in the 2007 KF2 South Garda Winter Cup.
António graduated to Formula Renault UK Winter Series with CR Scuderia for 4 races in 2008 as a guest where he proceeded to take a single podium. He also joined Motopark Academy full time for the Formula Renault 2.0 NEC where he took 9 podiums and a further win to finish runner up, the Eurocup Formula Renault 2.0 where he finished 13th and the Formula Renault 2.0 Portugal Winter Series where he finished 17th. He returned with Motopark for 2009 where he returned back to Formula Renault 2.0 where he took 11 races, of which 9 were wins, to win the championship and the Eurocup Formula Renault 2.0 where he took 6 podiums and 3 wins to finish 3rd in the standings. He went on to the Formula 3 Euro Series in 2010 with the team where he finished seventh with 3 wins and a further 1 podium and the Masters of Formula 3 where he placed 18th. In 2010, da Costa also joined Carlin for the GP3 Series for four races where he finished 26th and the Macau Grand Prix where he finished 6th.
He went on to compete full time in the GP3 Series with Status Grand Prix in 2011 where he went on to take a single race win to finish 13th in the standings. Da Costa also proceeded to join Hitech Racing for the British Formula 3 Championship and Macau Grand Prix. He took 3 podiums in the championship to place 13th while he did not finish in Macau. To finish off his 2011, da Costa joined Ocean Racing Technology for the GP2 Final where he finished 9th. António made his return to the GP3 with Carlin where he went to take 3 wins and 3 more podiums to finish 3rd in the standings. He also competed with the team for two races at the MotorSport Vision Formula Three Cup where he won both but wasn’t eligible to score points in the championship. He also returned to Macau with the team where he won the race. In that same year, da Costa raced in the Formula Renault 3.5 Series with Arden Caterham to take 6 podiums, of which 4 were wins, to finish 4th in the standings. He proceeded to focus completely on the 3.5 Series with Arden for 2013 where he took 3 wins and 3 further podiums to finish 3rd in the standings.
Da Costa debuted in the Deutsche Tourenwagen Masters with BMW Team MTEK in 2014. He finished 21st in the stands with 6 points. He made his return with BMW Team Schnitzer in 2015 where he managed to take a single win and 2 more podiums to finish 11th. António returned with the team in 2016 but only managed one podium and 17th in the standings. He also started his Stock Car Brazil series in 2015 with Full Time Sports but drove as a guest in his rookie year so wasn’t eligible for points in the championship. He returned with the team in 2016 where the same principles applied. He went on to join HERO Motorsport for 2017 and 2018 where he took a podium in each year but was a guest driver. da Costa also returned to Macau Grand Prix with Carlin in 2016 where he won the race once more.
His debut in the Blancpain GT Series Sprint Cup happened in 2017 where he joined Rowe Racing for 6 races. He proceeded to finish 26th in the standings in his rookie year. Alongside this, he also debuted in the International GT Open with BMW Team Teo Martin where he took a single win and another podiums to finish 12th in the standings. António made his way to WeatherTech SportsCar Championship in 2018 where he joining Jackie Chan DCR JOTA to finish 51st. In that same year, he joined BMW Team MTEK in the LMGTE Pro 24 Hours of Le Mans but he didn’t manage to finish the race. He repeated the Le Mans series in 2019 where he went on to finish 10th. He also debuted in the LMGTE Pro FIA World Endurance Championship with the team between 2018 and 2019 where he took one podium and finished 10th in the standings. António currently competed in the LMP2 FIA World Endurance Championship with Jota Sport for the 2019 to 2020 season where he currently sits at 4th with 2 podiums and a win.
António debuted in Formula One in 2010 where he joined Force India as a test driver. From there, he then went on to join Red Bull Racing in 2012 as a test driver for that year.
Da Costa debuted in the FIA Formula E during the 2014 to 2015 season where he joined Amlin Aguri where he took a single race win and finished 8th in the standings. He stuck with Aguri for the 2015 to 2016 season but he didn’t manage to get a podium and finished 13th in the standings. For 2016 to 2017, he remained with MS Amlin Andretti where he finished 20th with no podiums. The team made the name change in 2017 to 2018 when they became MS&AD Andretti Formula E, keeping Antonio as a part of the team. He placed 15th in the standings. From there, da Costa stayed with BMW i Andretti Motorsport for one last year in the 2018 to 2019 season where he took his first win since his debut and took 3 further podiums to finish 6th.