In a race overshadowed by the death of Moto3 rider Jason Dupasquier after a crash in qualifying on Saturday, Quartararo became the first rider to win in Mugello after starting from pole position since 2014.
Meanwhile, Marquez's hopes of securing a strong result in his fourth race since returning from a broken arm were ended at turn three on the second lap when he made contact with Brad Binder.
Quartararo started from his fourth consecutive pole, but he was usurped by Francesco Bagnaia almost immediately.
Ducati rider Bagnaia fell from the front on just the second lap, though, handing the advantage back to Quartararo, who fought off the threat of fellow Frenchman Johann Zarco to seal a third win in six races.
This latest victory extended his lead at the head of the standings to 24 points over Zarco, who was leapfrogged by Miguel Oliveria into second with eight laps remaining, with Joan Mir pushing him off the podium soon after.
Alex Rins pushed Zarco further down the standings, but he was denied a push at a podium place when he crashed on the final corner.
Oliveira beat Mir to the line for second, although the Portuguese rider was initially demoted to third for exceeding track limits on the final lap. It later transpired Mir had also committed the same offence and they were returned to their original positions.
1. Fabio Quartararo (Monster Yamaha)
2. Miguel Oliveira (Red Bull KTM) +2.592s
3. Joan Mir (Suzuki Ecstar) +3.000s
4. Johann Zarco (Pramac Ducati) +3.535s
5. Brad Binder (Red Bull KTM) +4.903s
6. Jack Miller (Ducati Team) +6.233s
7. Aleix Espargaro (Aprilia Gresini) +8.030s
8. Maverick Vinales (Monster Yamaha) +17.239s
9. Danilo Petrucci (KTM Tech3) +23.296s
10. Valentino Rossi (Petronas Yamaha) +25.146s
1. Fabio Quartararo (Monster Energy Yamaha) 105
2. Johann Zarco (Pramac Racing) 81
3. Francesco Bagnaia (Ducati Lenevo) 79
4. Jack Miller (Ducati Lenevo) 74
5. Joan Mir (Suzuki Ecstar) 65
1. Monster Energy Yamaha 169
2. Ducati 153
3. Pramac Racing 102
4. Suzuki Ecstar 88
5. KTM Factory Racing 64
The Barcelona-Catalunya circuit plays host to the next race in the championship next weekend, with Quartararo looking for a repeat performance of his win there last year.