Piastri finished second in the Italian Grand Prix after a bold strategy call from Ferrari left Charles Leclerc with track position.
However, Piastri had controlled affairs up until his final stop, beginning with a strong move on Norris on Lap 1.
The McLaren duo started side-by-side on the front row, with the Australian drawing alongside under braking for the second chicane.
As Norris defended the left-hander, Piastri moved around the outside, leaving him with the apex for the right-hander that immediately followed.
“Time and time again, he has shown that he’s ready to race at the front against the big boys,” Webber, who manages Piastri, said of the 23-year-old on the Formula For Success podcast.
“He’s fine for that: respectful, hard, clean.
“We saw again that, in the weekend with Turn 4 in Monza which is, to get through there and be absolutely brilliant on the brakes and get the move done as well as he did, is another feather in his cap.”
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Piastri sits fourth in the drivers’ championship, over 100 points down on leader Max Verstappen though mathematically still in contention for the title.
However, Norris is a closer and stronger bet, despite scoring fewer points than Piastri during F1’s European leg.
It has created a fascinating dynamic within McLaren with opinions split on how to best manage its competitive driver pairing for both short- and long-term success.
In Monza, the pair were free to race without team orders, though ‘Papaya Rules’ were in play – essentially acknowledgment that, while free to battle with one another, they must not make contact.
“His way of racing, week in, week out, his damage bill is incredibly low,” Webber said of Piastri.
“I think the respect from the other drivers that he’s raced against through the junior formulas has always been there, and he’s getting that respect.
“He wants to get respect, he doesn’t want to have people all applaud us – as we are doing now – he just wants to earn his stripes, go out there and do his talking on the track.
“That’s what he’s done so well in the last five years.”
The 2024 F1 season has eight events left, the next in Azerbaijan starting on Friday.