A native of Monterrey in Nuevo León, Suárez began his racing career in karting in 2002. In 2007 he won the class championship. In 2008 he moved to the preliminary category of NASCAR Mexico, Mini-Stocks, where he became the youngest driver to win a race in the series. He moved to the NASCAR Mexico Series in 2010, driving for Telcel Racing; despite running only a partial season, he won the series' Rookie of the Year title. In 2011, Suárez participated in the Toyota All-Star Showdown at Irwindale Speedway, finishing in 11th, the highest-finishing Mexican driver in the event. In the 2011 NASCAR Mexico Series, he finished the season with three poles and a podium, finishing ninth in the standings. At the same time, he participated in 7 races of the NASCAR K&N Pro Series East.
In 2012, he alternated his time between the Mexico Series and K&N Pro Series East. In Mexico, he led the points for most of the season and entered the final race of the year in contention for the championship, but wound up finishing third, having scored two victories. In the K&N Pro Series East, he finished in sixteenth place in the overall standings, achieving 3 top-10s in nine races.
Suárez competed for the full season in the K&N Pro Series East in 2013, driving a Toyota for Rev Racing. He scored his first victory in the series at Columbus Motor Speedway, also recorded six top-5 and nine top-ten finishes on his way to third in the championship standings. Meanwhile, he finished runner-up of the NASCAR Toyota Series achieved 3 wins and five podium finishes in the season. Suárez was also named to NASCAR's Drive for Diversity program during the 2013 season.
In 2014, Suárez returned to the K&N East and Toyota Series, winning the first two K&N East and first Toyota Series races of the season. In April, he was selected by Joe Gibbs Racing to make his debut in the Nationwide Series at Richmond International Raceway, driving the team's No. 20 Toyota; he finished 19th.
In August 2014, it was announced that Suárez would compete full-time in the 2015 NASCAR Xfinity Series in the No. 18 Toyota for Joe Gibbs Racing, and that he would also run a partial schedule in the 2015 NASCAR Camping World Truck Series for Kyle Busch Motorsports.
On July 4, 2015, Suárez won the pole for the Xfinity Series Subway Firecracker 250 at Daytona, his first career pole in the series. Suarez would go on to win 2 more poles at Iowa and Kentucky, and won his first ARCA pole at Kansas. Suarez went on to win the 2015 NASCAR Xfinity series rookie of the year title edging Darrell Wallace Jr. for the award by one single top ten finish in statistics.
In 2016, Suárez's car number was switched to No. 19. He won his first race at Michigan, by passing Kyle Busch on the last lap, becoming the first Mexican-born driver to win in a NASCAR national touring series.