It was the England-based team's fourth consecutive 1-2 this season.
Norris held off Piastri on fresher tyres to take the chequered flag by less than a second.
"I'm dead. It was tough," said the Brit after the race.
"I wasn't planning the one-stop strategy... my voice has gone, but even more rewarding because of that."
Norris' win takes him to within nine points of Piastri in the drivers' championship, ahead of F1's mid-season break.
"I pushed as hard as I could... looking forward to a few weeks off," said the Aussie after the race.
Polesitter Charles Leclerc, of Ferrari, held onto third for many of the later laps, but was pipped to the podium by Russell.
Fernando Alonso secured fifth for Aston Martin, ahead of Gabriel Bortoleto (Sauber), Lance Stroll in the other Aston, and Liam Lawson (Racing Bulls).
Max Verstappen, in his 200th drive for Red Bull, had to settle for ninth, with Mercedes rookie Andrea Kimi Antonelli rounding out the top 10.