The 25-year-old Briton, who trails by 31 points in the drivers' title race, clocked a best lap in one minute and 42.704 to top the times, beating his 24-year-old Australian rival by 0.310 seconds.
A similar 1-2 outcome in Sunday's race would clinch a second consecutive constructors' championship success for McLaren and leave the duo to fight for the individual title without the spectre of team orders.
Ferrari's Charles Leclerc was third, half a second adrift, closely followed by fourth-placed George Russell of Mercedes.
Williams' Alex Albon was fifth, ahead of the Red Bulls of Yuki Tsunoda and four-time champion Max Verstappen, Williams' Carlos Sainz and the Racing Bulls pair of Liam Lawson and Isack Hadjar.
Kimi Antonelli, seeking a clean session after a series of inconsistent outings at the European races, was 11th for Mercedes, and seven-time champion Lewis Hamilton of Mercedes was down in 13th after a brush with the barriers.
The session began in warm sunshine with Franco Colapinto leading the way for Alpine ahead of the rest of the pack, including a sickly Russell who had missed Thursday's media sessions due to illness.
Verstappen set the early pace before being overhauled by Russell and Hadjar as the track improved, with the traffic - all running on new extra-soft tyres for the street circuit demands - clearing leaves and laying down rubber in rising temperatures.
Ten minutes in, Piastri pitted for attention to a possible engine problem while Leclerc worked his way to top the times before the action was red-flagged, due to debris at Turn 16, as Norris took over.
The debris came from bonding material used to retain the kerb, and repairs required a 24-minute pause before the session resumed with Piastri back in his car and 20 minutes remaining.
Norris, needing a run of good results to reduce Piastri's advantage in the title race, was showing good pace and improved his best lap to 1:42.704, a full second clear of Verstappen with 10 minutes to go and Piastri in third.
Ferrari's mixed fortunes continued as Leclerc rose to second in 1:43.286 on one of his favourite circuits, while Hamilton hit the wall at Turn Five, damaging his front wing and picking up a puncture.
Hamilton trundled back to the pits, but after rapid repairs rejoined the fray.