Quartararo set a time of 1m32.017, nearly half a second faster than his FP3 time, which was also a new lap record.
The Frenchman was a close second behind the Suzuki of Alex Rins after the first runs, but he made a big improvement after his mid-session pitstops.
First, it was Maverick Vinales demoting Rins, but Quartararo went faster than his factory counterpart by 0.140s.
Vinales will complete a Yamaha 1-2 on the grid with Rins settling for third, having improved by only 0.003s on his second run.
Marc Marquez will not start from the front row for the first time this year as he was fourth in qualifying.
He miraculously avoided a crash at Turn 1, and then followed Andrea Dovizioso on his final lap, but he got too close to the Italian and couldn’t improve.
Joan Mir brought the second Suzuki into the top five, despite only making it to Q2 after Valentino Rossi’s top time was deleted at the end of FP3.
LCR Honda’s Cal Crutchlow was sixth ahead of the top Ducati of Danilo Petrucci, who was 1.2s off the pace.
Crutchlow’s teammate Takaaki Nakagami was eighth ahead of Franco Morbidelli (SIC Yamaha) and Jack Miller (Pramac Ducati).
Dovizioso was a disappointing 11th, with KTM’s Pol Espargaro rounding out the Q2.
Rins controls Q1, Espargaro impresses
Rins dominated the first part of qualifying, leading from his first flying lap until the finish.
He had a six-tenth advantage over Francesco Bagnaia after his first run and then made a small improvement just before the chequered flag to end up with a 1m32.710s.
Bagnaia was eventually denied a Q2 spot by Pol Espargaro, who made a heroic effort to go through with an injured right wrist, a legacy of a crash in the Barcelona post-race test.
The Pramac Ducati rider will start 13th alongside his mentor Valentino Rossi, who missed out on Q2 by 0.166s.
He suffered his worst qualifying at Assen since 2006, when he was 18th.
The also injured Aleix Espargaro qualified 15th, beating his teammate Andrea Iannone, down in 20th, by four tenths.
KTM’s Johann Zarco was a disappointing 18th, having done his first run with a hard rear tyre, and then crashed at T5 after switching to a soft.
Tito Rabat had a tumble in FP4 and missed the first run, eventually qualifying last by six tenths.