Italian rider Morbidelli powered through the final sector at Circuit de Catalunya to deliver a scorching lap of one minute 38.798 seconds.
Early pace-setter Quartararo went for one final attempt as the chequered flag came out but was unable to improve on his second-place finish.
Italian great Valentino Rossi, who committed his future to Petronas Yamaha for the 2021 season earlier on Saturday, finished third on his factory Yamaha bike to secure a front row position for the first time this season.
Championship leader Andrea Dovizioso's difficult run in qualifying sessions this season continued as the Ducati rider failed to qualify for Q2. He will start Sunday's race 17th on the grid. (Reporting by Hardik Vyas in Bengaluru; editing by Clare Fallon)