Firman comes from a motorsport family: his father, Ralph Firman Sr., co-founded the Van Diemen racing car constructor. His sister Natasha is also a racing driver. Firman was educated at Gresham's School between 1988 and 1993 before moving directly into professional motorsport.
Firman was competitive immediately in British Formula Three. He narrowly missed the 1995 title despite leading much of the championship, losing it at the final round to Oliver Gavin. He returned in 1996 and captured the championship at his second attempt.
At the 1996 Macau Grand Prix, Firman claimed victory under unusual circumstances. In the second race he was leading before a broken front wing allowed Jarno Trulli to pass him on the final lap. Firman then crashed at the hairpin and blocked the circuit, causing the race to be stopped; under the regulations the result was declared at the end of the previous lap, when Firman had still been ahead, giving him the win.
Firman moved to Japan where he became a consistent front-runner. In 2002 he won the Formula Nippon Championship, the premier Japanese open-wheel series.
Firman secured a seat with the Jordan Formula One team for 2003 alongside Giancarlo Fisichella. He made his debut at the 2003 Australian Grand Prix and competed in 14 Grands Prix during the season. His only points finish came at the Spanish Grand Prix, where he scored one championship point — the first point for a Republic of Ireland-licensed driver since Derek Daly in 1982.
His season was interrupted by a serious crash during practice for the 2003 Hungarian Grand Prix, forcing him to miss that race and the following Italian Grand Prix. Zsolt Baumgartner replaced him for both. In November 2003, Firman drove a Jordan-Ford EJ13 at the Macau Grand Prix's 50th anniversary celebrations — the first time a contemporary Formula One car had lapped the tight Guia Circuit in competition — recording a 1:59.4 lap time, 13 seconds faster than the Formula Three pole position.
He was not retained by Jordan for 2004.
Firman continued to thrive in Japanese motorsport after his F1 stint. In 2007 he and co-driver Daisuke Ito won the Super GT Series GT500 class championship with the ARTA team, becoming champions before the final race of the season — a first in the series' history. He was also runner-up in the Super GT in 2002, 2005 and 2009. Firman competed in the 24 Hours of Le Mans and drove for A1 Team Ireland in the A1 Grand Prix series.
He retired from racing in 2013 and subsequently ran a British engineering company.