Russell appeared on course for a comfortable win as the race unfolded with few incidents of note in the opening stages.
Piastri moved up from third on the grid to second by passing Ferrari’s Charles Leclerc at the start but the Australian lacked the pace to chase Russell, who edged away into a comfortable lead.
Behind, Piastri soon found himself holding off Red Bull’s Isack Hadjar, who won an early battle with McLaren’s Lando Norris and then passed Leclerc to challenge the McLaren.
Meanwhile, Verstappen was making his way up from eighth place after a qualifying session in which a cascade of problems had prevented him qualifying second to Russell.
By lap seven, he was past Hadjar into third place after the Frenchman did not fight him.
The three cars circulated together in a stalemate, with no-one able to pass and the teams all waiting for the expected accident that would lead to a safety car and a cheap pit stop.
That happened on lap 30, when Williams’ Alex Albon crashed at Turn Five.
On the restart, six laps later, Verstappen passed Piastri into Turn One to move into second place.
But the race was almost immediately put back under caution after a pile-up at the first corner caused by Alpine’s Franco Colapinto, who cannoned into team-mate Pierre Gasly, who in turn clattered into Norris.
Behind them, the two Racing Bulls also came together, Arvid Lindblad tipping Liam Lawson into a spin, leading to a sarcastic response from the New Zealander on the radio.
That put Verstappen on to Russell’s tail for another restart but the Briton handled the situation coolly, accelerating early and catching the Dutchman somewhat unawares.
He was unable to shake Verstappen, who was always about 0.5secs behind but never quite close enough to try a move into Turn One, the only significant overtaking place.
There was one last chance for Verstappen on the final lap.
Valtteri Bottas crashed his Cadillac at Turn 15, and after backing off for the yellow flags, Russell lost his turbo which in turn cost him electrical boost.
Verstappen closed in on the run to the line and there was just 0.196secs between them as they crossed the line, the closest green-flag finish to a grand prix since the 2002 United States Grand Prix.