Costa was born in Parma in the Italian region of Emilia-Romagna. He attended the liceo scientifico in Parma before studying Mechanical Engineering at the University of Bologna, where he graduated in 1986 with a doctorate grade of 100 out of 100. His thesis examined Formula One car suspension systems using data gathered from the Ferrari racing department. By 1987 he had qualified as a professional engineer and joined the Register of Engineers of the Province of Parma.
Following his graduation, Costa joined the Fiat group through Abarth in Turin, where he worked in the calculation department on rally cars including the Lancia Delta Integrale. His work as a stress engineer attracted the attention of Giancarlo Minardi, who recruited him at the age of 27 to join the Minardi Formula One team.
Costa joined Minardi in early 1988 as chief car designer, initially focusing on the car's suspension alongside aerodynamicist Nigel Cowperthwaite. He rose to technical director by January 1991, overseeing all of Minardi's technical operations until August 1995. Under his leadership, Minardi achieved what remain the team's best-ever results: a historic front-row qualifying position for Pierluigi Martini at the 1990 United States Grand Prix, where Martini also set the second fastest race lap, and a best race finish of fourth place achieved multiple times between 1991 and 1993. These results helped Minardi secure a supply partnership with Ferrari, which delivered Ferrari engines to the team from the 1991 season onwards.
Costa joined Ferrari in September 1995, initially assigned to develop the grand touring Ferrari F50 GT for the BPR Global GT Series. By January 1996 he became head of the chassis design office, and in 1998 he was appointed assistant to chief designer Rory Byrne, working closely with both Byrne and technical director Ross Brawn.
The period from 1999 to 2008 represented Ferrari's greatest era of dominance in Formula One. The team won eight Constructors' Championships (six consecutively) and six Drivers' Championships (five consecutively with the same driver, a record that stood into the 2020s). Costa was one of the principal architects of that era's hardware, contributing to landmark cars including the F2002 and F2004 — widely regarded among the most effective Formula One cars ever built.
When Byrne announced his retirement in 2004, Costa was named his successor as chief designer, a role he held from July 2004 to December 2006. Although Byrne credited Costa with the design of the 2005 Ferrari and expressed confidence it would surpass the F2004, that season proved difficult: a mandatory rule change banning mid-race tyre changes neutralised Ferrari's tyre management advantage, limiting the team to a single win at the controversial 2005 United States Grand Prix. Costa was subsequently promoted to head of design and development before becoming chassis director in January 2007 and technical director from January 2008.
Costa's dismissal from Ferrari came after the 2011 Spanish Grand Prix following a poor start to the season. He later described his departure as handled in a manner he could not describe as elegant, noting that the F2007 and F2008 — the last Ferraris to win a world championship — were produced under his technical leadership. Costa argued that key strategic errors, including a decision not to proceed with a new wind tunnel he and his team had requested in 2008, contributed to the team's subsequent decline.
In September 2011 it was announced that Costa would join Mercedes as engineering director. He led the design of the 2013 Mercedes car under technical director Bob Bell before taking charge of the team's championship-winning hardware through the hybrid era. The Mercedes cars he oversaw won the Constructors' and Drivers' Championships in 2014, 2015, 2016, 2017, and 2018, creating a dynasty that eclipsed even the Ferrari years Costa had helped build.
Costa described the parallel between Michael Schumacher and Lewis Hamilton — the two champions he worked most closely with — as greater than might be assumed, characterising both as united by an obsessive pursuit of perfection, though distinct in their methods of engagement with technical staff. He noted that Hamilton would call him at home on Sundays to discuss car details, while Schumacher's focus was on when a specific improvement could be tested on track.
In July 2018 Costa stepped down as engineering director to become a technical consultant for the 2019 season, citing the desire to spend more time with his family in Italy. He departed Mercedes entirely in September 2019.
On 6 September 2019, Costa was announced as the incoming chief technical officer of Dallara, succeeding company founder Giampaolo Dallara. He took up the role formally in January 2020. At Dallara, Costa oversees an engineering portfolio spanning seventeen automotive projects including IndyCar, Formula E, and a collaboration with Ferrari on the Le Mans Hypercar programme.
In October 2022, Costa was appointed an Officer of the Order of Merit of the Italian Republic. Among all chief designers in Formula One history as analysed by a 2022 study, Costa was ranked among the most successful Italian designers alongside Mauro Forghieri. His championship-winning cars achieved 168 wins from 257 Grands Prix — a strike rate of 65.4%.