The Australian lapped the Lusail desert circuit with a best time of one minute 20.924 seconds, with championship leader Norris 0.058 seconds slower and running wide at the final corner late on.
Aston Martin's Fernando Alonso was third with Williams' Carlos Sainz fourth and Racing Bulls' Isack Hadjar fifth fastest.
Red Bull's four-time world champion Max Verstappen, the third title contender and a two-time winner in Qatar, was sixth and complaining about his car 'jumping' and the steering being 'weird'.
Norris leads Piastri and Verstappen by 24 points with two rounds remaining, and the Briton could seal his first championship on Sunday if results go his way.
Friday's session was the sole practice because Qatar is a sprint weekend, with qualifying for that 100km 25-lap Saturday race taking the place of the usual second practice.
Williams had Alex Albon in seventh place with Ferrari's Charles Leclerc eighth, Aston Martin's Lance Stroll ninth, and Mercedes' Kimi Antonelli 10th.
"I cannot feel a thing. Really frustrating, I don't even know where the balance of the car with this power steering," said Leclerc, whose teammate Lewis Hamilton was 12th fastest, over team radio.