Their battle had allowed Norris to close up to them, after wasting little time passing the drivers between him and the leaders.
Norris passed Verstappen by diving around the outside into Turn 11 on lap 17, but he ran off track in doing it, and had to give the place back.
But a lap later, he made the move with ease, Verstappen this time appearing not to bother defending after sensing the game was up.
Verstappen quickly dropped back from the McLarens, losing nine seconds in as many laps before his pit stop.
The world champion’s day became worse when the virtual safety car (VSC) was deployed during the pit-stop period.
Russell, who was the only driver in the top 10 to start on the hard tyres like Hamilton, had not yet stopped and that VSC allowed him to pit for fresh tyres and rejoin still ahead of Verstappen.
For a while, Verstappen closed in on Russell but he could get no closer than about 1.8 seconds before failing to make any further progress.
Norris also closed on Piastri at the front, but after halving his deficit to four seconds, the Australian upped his pace to show he had time in hand and led comfortably to the end.
It is the first time a McLaren driver has won three races in a row since Mika Hakkinen over the 1997-98 seasons.
Piastri’s win extends his championship lead over Norris to 16 points, with Verstappen the same margin behind in third place.
