Piastri clocked a lap record time of one minute 8.662 seconds, 0.012 seconds ahead of Norris.
It was the fifth pole position of the season for the Australian driver, who is seeking to extend his nine-point world championship lead over Norris in Sunday's race.
Norris had had the edge over the Australian in all three practice sessions so far, but Piastri hit back when it counted to give himself an advantage for the race