Coming from a racing family, da Costa started karting at the age of nine, racing in a number of Portuguese championships against more experienced and older competitors. In 2007, he progressed into single-seaters in Formula Renault 2.0 and, after gaining attention in both Formula 3 Euro Series and GP3, he would eventually join the Red Bull programme. After making his Formula One test debut for Force India in 2010, the Portuguese driver went on to pilot four of Adrian Newey's title-winning Red Bull Racing machines.