Muñoz began motorsport in karting in 2002 and spent several seasons developing his craft before transitioning to single-seaters in 2007. That year he contested the Formula TR Pro Series, taking one podium in four races and finishing eighth overall.
In 2008 he stepped up to Formula Renault, racing for Prema Powerteam in both the Italian Formula Renault 2.0 and Eurocup Formula Renault 2.0. His Italian campaign yielded six points-scoring finishes in twelve races; his Eurocup season was more modest, with a best finish of thirteenth. For 2009, now with Epsilon Euskadi, Muñoz improved significantly in the Eurocup, scoring ten points-scoring finishes in fourteen races — including a podium at the Nürburgring — to finish eighth in the standings. He also competed in the Formula Renault 2.0 West European Cup that year, collecting two podium finishes and ending seventh in the championship.
Muñoz broadened his single-seater experience by contesting three rounds of the European F3 Open Championship in 2009 with Porteiro Motorsport, his best result a second place at Spa-Francorchamps. He moved to the Formula 3 Euro Series full-time in 2010, joining Mücke Motorsport alongside Roberto Merhi. A runner-up finish at Brands Hatch was his highlight as he finished ninth in the championship. He also made six guest appearances in the Invitational Class of the British Formula 3 Championship that year.
Muñoz returned to the Formula 3 Euro Series in 2011 with Signature, achieving a best finish of third at Silverstone and ending the season eighth overall.
In 2012, Muñoz relocated to the United States to join Andretti Autosport in Indy Lights. He won twice — on the oval at Auto Club Speedway and from pole position in Edmonton — and added three further podiums to finish fifth in the championship. He returned to Indy Lights for 2013 with Andretti before being given the opportunity to contest the Indianapolis 500 with the same team.
Muñoz qualified on the front row for the 2013 Indianapolis 500, starting second. He ran at the front throughout the race, led twelve laps, and finished second, earning Rookie of the Year honors. Later that summer he made a brief cameo with Panther Racing at the Honda Indy Toronto, replacing the injured Ryan Briscoe for Race 2 after just twelve laps of morning warmup and finishing seventeenth from twenty-fourth on the grid.
He remained a full-time IndyCar competitor with Andretti Autosport in subsequent seasons. His first IndyCar victory came on 30 May 2015, in the rain-shortened Race 1 of the Detroit Belle Isle Dual Grand Prix. The 2016 Indianapolis 500 brought his second runner-up finish at the Brickyard, this time behind teammate Alexander Rossi. After the 2016 season, Muñoz departed Andretti Autosport.
For 2017 he joined A. J. Foyt Enterprises, but the partnership lasted only one season. In 2018, Muñoz made a one-off return to the Indianapolis 500 with Andretti, finishing seventh. He closed his IndyCar career that year by filling in for the injured Robert Wickens at Schmidt Peterson Motorsports for the final two rounds of the season at Portland International Raceway — where he started fourteenth and finished twelfth — and at Sonoma Raceway.
Muñoz's two Indianapolis 500 podiums, earned in his debut and then three years later, mark him as one of the most prominent Colombian open-wheel racers of his generation. His transition from European junior formulae to the top level of American open-wheel racing, culminating in a victory at Detroit in 2015, illustrated the pathway that IndyCar increasingly offered to international talent in the 2010s.
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