Piastri took the lead at the start as he launched off the line better than Norris and got down the inside at Turn One.
They went three-wide around the corner with Verstappen on the outside, and the Dutchman then swept into the run-off area, responding to Piastri jinking slightly left and Norris doing the same, and used a slingshot to pass Norris as he returned to the track.
Norris complained that Verstappen had broken the rules and had to give the place back. Verstappen disagreed but after the stewards said the incident was under investigation, Red Bull advised their driver to cede the place.
Verstappen did so, but responded with the first of a long series of angry messages.
At the front, Piastri seemed in control of the race, extending a lead of more than three seconds before his first stop and then building it to nearly five by lap 32, shortly before half-distance.
But then he ran wide at the fast Turn 11, the lead immediately came down to 2.1 seconds, and Norris began to close in.
Norris held the gap at less than two seconds for 10 laps, until McLaren decided to pit the Briton to ensure he was not ‘undercut’ by Hamilton, who had made his final stop on lap 40.
The risk for McLaren was that Hamilton would close in enough on fresh tyres that Norris would come out behind him when he made his own stop.
So McLaren pitted Norris to protect against this, and then stopped Piastri two laps later.
But this meant that Norris had gained time on his fresh tyres and when Piastri rejoined, he was three seconds behind his team-mate.
The bargaining over who would win the race continued for the next 20 laps until Norris eventually decided his best long-term interests were in doing what he was told.
Behind Verstappen, Ferrari’s Carlos Sainz took sixth, while the second Red Bull of Sergio Perez and Mercedes’ George Russell recovered from their starting positions close to the back to take seventh and eighth.
RB’s Yuki Tsunoda and Aston Martin’s Lance Stroll completed the points positions in ninth and 10th.