Piastri ran second for much of the opening stint and was in a strong position to record a top-three finish following his pit stop.
However, a Safety Car sooner after his stop put paid to his chances, while Norris was ideally placed to benefit from the interruption.
Yet to pit, the Brit swapped onto the hard compound tyres while the race was neutralised to maintain the lead he gained courtesy of running long in the opening stint.
Piastri took the restart fourth and quickly came under pressure from Carlos Sainz behind.
An all-mighty scrap then developed, the Australian seeing off the Ferrari’s challenge at Turn 11 on two occasions.
He eventually lost out at Turn 17 when the Spaniard slithered up the inside, making contact on his way through. After that, Piastri fell back two more places as he battled front-wing damage.
That forced him back into the pits for a replacement, ruining any chance he had of a strong result, though he did claim fastest lap as he charged back to be classified 13th at the chequered flag.
Officials confirmed they would investigate the clash with Sainz after the race, Piastri refusing to be drawn on the Turn 17 contact.
“I need to see,” he responded when asked if Sainz was overly aggressive.
“I mean, we weren’t particularly close to the apex, but I need to see it first.”
Meanwhile, Norris went on to win the race after he was never headed following the restart.
Piastri was without the full upgrade package fitted to Norris’ car in Miami, with suggestions that left him at a two-tenth per lap deficit over his team-mate.
“Definitely a lot of positives to come from today,” Piastri reasoned.
“For both of us, the car was really strong.
“For Lando, to win the race on pace and to pull away from Max [Verstappen] after the Safety Car, I think that’s a very, very encouraging sign.
“So very happy for him and for the whole team. I think we deserve it.
“Our trajectory in the last 12 months has been towards this moment.
“Even with the car that I had, I think we were very strong as well,” he added.
“Excited for Imola.”
The final result belied a strong opening to the race, which saw Piastri rise to second in the early laps.
He then kept Max Verstappen in his sights before his stop, such that Red Bull Racing instructed the Dutchman to try and extend his lead to cover off the threat the McLaren driver posed in the pit cycle.
“I was happy with the start,” Piastri said.
“The fact that I could get past Leclerc as well was unexpected coming into the race.
“From that point, I was quite happy and was reasonably comfortable with him behind me there, just timing of the Safety Car wasn’t ideal.”
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