Red Bull's reigning champion Max Verstappen will join Piastri on the front row for the first race since Christian Horner was dismissed as team boss and replaced with Laurent Mekies.
Piastri lapped the Spa-Francorchamps circuit with a best time of one minute 40.510 seconds, 0.477 seconds quicker than Verstappen and 0.618 clear of Norris.
The Australian is eight points clear at the top after 12 of 24 rounds.
Ferrari's Charles Leclerc starts fourth, but teammate Lewis Hamilton will line up 18th after a difficult afternoon for the seven-time world champion, whose most recent win came at the same circuit last year with Mercedes.
The Briton spun on his last flying lap while on course to go through, with the suspicion falling on a failure of the car's rear axle.
George Russell, who finished first last year for Mercedes but was then disqualified for an underweight car, also struggled and qualified 13th.