Quartararo, on a satellite Yamaha, was followed home by Spaniard Maverick Vinales and Italian veteran Valentino Rossi, both on Yamaha factory bikes.
With the temperature reaching 40 degree Celsius and estimated at 60 degrees on the track, only 13 of the 21 starters finished.
“That was tough,” said the 21-year-old at the finish.
He has now won both races in the coronavirus-delayed MotoGP season, with last week’s Spanish Grand Prix success the first by a French rider in MotoGP since 1999.
Quartararo has now won more races at the elite level than he did in four seasons in Moto2, where he recorded one victory, and Moto3, where he twice made the podium but never won.
“It feels so good to make back to back wins,” he said.
Quartararo has a 50-point lead over the reigning champion Marc Marquez, who was absent with a broken arm after his crash in the season opener.
Vinales is second on 40 points after finishing runner up for the second straight race.
Vinales rued a mistake on the first lap.
“I tried to overtake Fabio but then I went wide and Fabio and Valentino overtook me,” he said.
As Quartararo burst clear, Rossi, who started on the second row of the grid, jumped to second, overtaking Vinales, who said that he had struggled to cope in his teammate’s exhaust.
“It was impossible to breath,” Vinales said. “I was stuck behind Valentino and I could not breath. I was destroyed. Maybe the heat from the other bikes.”
While Quartararo disappeared ahead in the heat haze, the two Yamaha works riders had to fend off challenges from the Ducati-Pramac pair of Italian Francesco Bagnaia, who climbed to second before his engine failed, and Australian Jack Miller, who crashed.
Japanese rider Takaaki Nakagami (Honda-LCR) inherited fourth place.
– ‘He’s my idol’ –
Rossi showed that, at 41, he is still capable of competing. The seven-time MotoGP world champion has always enjoyed cheeky celebrations and this time climbed onto a barrier and punched the air in a victory dance in front of a grandstand completely empty due to COVID-19 restrictions.
“To come back on the podium is not like a victory, but similar,” he said.
Quartararo, who makes no secret of his admiration for Rossi, agreed.
“He’s my idol and always has been,” said the young Frenchman. “It’s fantastic to be next to him on the podium for my second win.”
As he took the chequered flag, Quartararo also nodded to another sporting hero crossing his arms in an imitation of France striker Kylian Mbappe’s goal celebration.
Paris Saint-Germain’s Mbappe, who is nursing an injured ankle, responded on Twitter: “Congratulations. We can say that you have good taste… 10/10.”
Briton Cal Crutchlow, who returned to the track on a misfiring bike when it became clear that every finisher would collect points and limped in more than a lap behind to pocket three of them.
With Marquez missing, his younger brother and Honda teammate Alex, on only his second start at this level, set off 21st and last on the grid after a crash in qualifying but climbed to eighth by the end.
Earlier, Enea Bastianini gained his first Moto2 victory when he beat fellow Italian Luca Marini.
Japanese rider Tetsuta Nagashima, who crashed in practice, could only manage 11th, but kept the championship lead, two points ahead of Bastianini.
Marco Bezzechi, who started from pole position, finished third, completing an all-Italian podium.
In Moto3, Japan’s Tatsuki Suzuki won as championship leader Albert Arenas crashed on lap 15.
Suzuki, who started from pole, scored his second victory in the class and climbed to second in the championship behind Spaniard Arenas who had won the first two races of the season.