Moving into the Moto3 class of the CEV Repsol series in 2013, Quartararo joined Wild Wolf Racing – run by former Grand Prix racer Juan Borja – riding a Honda. Quartararo finished on the podium in his maiden race in the series, run in wet conditions, finishing second to Great Britain's Wayne Ryan at Circuit de Barcelona-Catalunya. Quartararo finished sixth in the second race at the circuit, and left tied for the championship lead with Dutch rider Bryan Schouten. Over the next four races, Quartararo recorded only one top-ten finish – from pole position at Navarra – and had dropped to eighth in the riders' championship standings, 37 points behind Spain's Marcos Ramírez. Quartararo finished the season strongly however, winning each of the final three races from pole position – his first series wins – defeating Ramírez by almost ten seconds in the final race at Jerez. As a result, Quartararo became the first non-Spanish rider since Stefan Bradl in 2007 to take the title, and at the age of 14 years, 218 days, its youngest series champion, surpassing the previous record held by Aleix Espargaró.

Quartararo remained in the CEV Repsol championship for the 2014 season, as he was not old enough to graduate to the World Championship level. The rule, announced in 2008 and introduced in 2010, stated that a rider must be 16 years of age to compete in a Grand Prix. Quartararo continued riding a Honda in the series, but moved to the Estrella Galicia 0,0 junior team run by Emilio Alzamora, the 1999 125cc world champion. Quartararo finished the season as a clear champion, winning nine of the season's eleven races, and finishing second – to María Herrera at Jerez and Jorge Navarro at Albacete – in the other two. His eventual championship-winning margin was 127 points over Navarro, who joined him as his team-mate at the final round of the season in Valencia. At that final round, Quartararo also beat World Championship competitors Alexis Masbou and John McPhee, with their SaxoPrint-RTG team making a one-off appearance in the championship.

Quartararo's performances in the Spanish series were noted at World Championship level. In the race which supported the 2014 French Grand Prix at Le Mans, Quartararo won by almost four seconds over nine laps, leading almost the entire race having started from second on the grid. In August 2014, the Grand Prix Commission – consisting of representatives from Dorna Sports, the Fédération Internationale de Motocyclisme (FIM), the International Road-Racing Teams Association (IRTA) and the Motorcycle Sports Manufacturers' Association (MSMA) – announced a change to the previously-introduced age eligibility rules, allowing for the champion of the FIM CEV Moto3 championship (regardless of age) to compete in the succeeding season of the Moto3 World Championship.

At his opening race weekend in Qatar, Quartararo qualified on the second row of the grid in sixth position, just 0.123 seconds away from the pole-setting time recorded by countryman Alexis Masbou. In the race, Quartararo was ever-present in the lead group, and held the lead of the race with two laps to go, but contact with Francesco Bagnaia saw both riders slip down the running order, with Quartararo ultimately finishing the race in seventh position. Such was the close nature of the racing that Quartararo was just 0.772 seconds behind the race winner, Masbou. At the following event in Austin, Texas, Quartararo achieved his first podium finish, with a second-place finish behind Danny Kent. He achieved his first pole position at the Spanish Grand Prix, a tenth of a second clear of Kent, but finished the race in fourth place. On home soil at Le Mans, Quartararo again took pole position, by just over a tenth of a second from team-mate Navarro. He led for a period during the race, but ultimately high-sided out of it from fourth position.

Quartararo returned to the podium with a second-place finish at Assen, having been part of the lead group for the entire race; he finished 0.066 seconds behind race winner Miguel Oliveira. In the next four races, Quartararo alternated retirements with finishes of eleventh at Indianapolis and fourth at Silverstone. At Misano, Quartararo crashed during the second free practice session, fracturing his right ankle – missing the race as a result of the injury. Quartararo also missed the following race in Aragon, where he was replaced by the team's CEV Moto3 rider Sena Yamada. Quartararo returned for the Japanese Grand Prix, but withdrew from the race after qualifying 29th on the grid, due to continued pain. Quartararo also pulled out of the Australian Grand Prix, after qualifying 19th on the grid. Quartararo ultimately finished the season in tenth place, with ninety-two points.

On 26 September 2015, it was announced that Quartararo would leave the Estrella Galicia 0,0 team to join Leopard Racing on a two-year contract from the 2016 season. Quartararo was considered as the title favorite for the season because of his performance in his rookie year and also because of the Leopard Racing team which won the previous year title with Danny Kent. But the season become a disastrous year for Quartararo. In the first three races Quartararo finished in thirteenth place followed by a crash in Spain. Quartararo finished sixth at his home race at Le Mans. Quartararo went pointless in six races and his best result being a fourth place finish at Austria.

Quartararo changed from Moto3 to Moto2 in 2017 to the Pons Racing as a teammate of Edgar Pons. On his debut, Quartararo finished 7th in Qatar. His only better result of the season was 6th place in San Marino. He finished the season with 64 points in 13th place of the championship.

For 2018, Quartararo switched to the Speed Up Racing team. He achieved his first-ever Grand Prix victory in Catalunya. Despite success in the following race with a 2nd place in Assen, the remainder of his season was difficult, achieving no further podiums and finishing with 138 points in 10th place of the championship.

In August 2018, it was announced that Quartararo would join Franco Morbidelli at the newly-created Yamaha satellite team, Petronas Yamaha SRT, in 2019.

He qualified on pole position for the Spanish Grand Prix. With this pole position he took the record for youngest ever polesitter in the MotoGP class, a record previously held by Marc Marquez since 2013. He also qualified on pole position in Catalunya and the following race in Assen, setting a new lap record and becoming the youngest rider with consecutive pole positions in MotoGP history. He finished third in Assen, Behind Maverick Viñales and Marc Marquez. The next race at the Sachsenring, he qualified second, behind Marc Marquez. During the second lap of the race Quartararo slumped in the third corner while sitting on the inside with Danilo Petrucci. He ended up in the gravel. This crash was the first crash during a race in his rookie season in MotoGP.