Australian Piastri, winner of four of the last five races, lapped the Imola circuit with a best time of one minute 16.545 seconds in the opening session and then lowered that to 1:15.293 in the later one.
Norris, 16 points behind his championship-leading teammate in the standings after six races, was 0.032 and 0.025 slower in the respective practices.
Imola, an old-style circuit with a tragic past as the track where Ayrton Senna died in 1994 and also expected to drop off the calendar next year, is Ferrari's home race and marks the start of the European season.
The first session was red-flagged and did not restart after Brazilian rookie Gabriel Bortoleto crashed with some three minutes remaining. He stepped out unhurt.
Carlos Sainz was third in that session for Williams, 0.052 off the pace, with Mercedes' George Russell a further 0.002 slower.
Pierre Gasly was a surprise third for Renault-owned Alpine in Practice Two, but 0.276 seconds slower than Piastri, with Russell fourth.
Racing Bulls rookie Isack Hadjar spun off at speed in the second session, and his car got stuck in the gravel, triggering more red flags.
Seven-time world champion Lewis Hamilton, limbering up for his first race in Italy as a Ferrari driver, was fifth and 11th respectively - 0.096 and 0.650 off the pace.
Red Bull's four-time world champion Max Verstappen was seventh and fifth, with Ferrari's Charles Leclerc ending the day sixth fastest.
"I can't rely on the rear. It feels like I'm drifting everywhere," complained Verstappen, last year's winner at Imola, over the radio during first practice.
Both Ferrari drivers complained about their brakes.
Mercedes' 18-year-old rookie Kimi Antonelli, the only Italian on the starting grid and preparing for his home debut, was 13th and 18th fastest.
Argentine rookie Franco Colapinto, making his debut for Alpine after replacing dropped Australian Jack Doohan, lapped 17th fastest in practice one but was 13th by the end of the afternoon and stayed out of trouble.
Leclerc, who was unwell on Thursday and skipped his media duties, was 12th in session one when he also ran wide into the gravel. He complained that his helmet was lifting.
