The race started on a wet track after a soaking wet morning, but with the sun out and more rain heading towards Silverstone.
Verstappen held the lead through a brief challenge from Piastri off the line but he could not shake the McLarens, and Piastri was soon challenging him hard for the lead.
Before he could try a move, though, a virtual safety car was deployed after Sauber's Gabriel Bortoleto crashed at Turn Two on lap four.
The race was restarted on lap seven, and Piastri was past Verstappen on the Hangar Straight before two laps were over, building a 2.9-second lead after just one lap.
It was then Norris' turn to challenge the Red Bull, but Verstappen gifted him the position when he slid off at Becketts on lap 11, just as heavy rain started.
That brought the drivers into the pits for fresh intermediates. By then, Aston Martin's Lance Stroll had gained time by an earlier change to soft tyres when the track was drying and emerged in fourth place, with Hulkenberg in fifth after stopping for inters on lap 10, so benefiting when others had to drive on a wet track on slick tyres and then pit.
The rain became heavier and a safety car was deployed on lap 14. The race restarted on lap 18, but a second safety car was sent out within a lap after Racing Bulls' Isack Hadjar ran into the back of Kimi Antonelli's Mercedes at Copse because he could not see him in the spray.
When the race restarted for the last time on lap 22, Piastri started to try to build a lead but he was not able to get more than four seconds ahead before Norris came back at him - he was less than two seconds back by the time Piastri pitted to serve his penalty with nine laps to go.
Stroll was unable to hold on to his third place, soon being passed by Hulkenberg, then Hamilton, and eventually dropping down to seventh place.
Behind Verstappen, Alpine's Pierre Gasly, who was fifth at the restart, lost places to Hamilton and the Dutchman but was able to secure sixth place.
Williams' Alex Albon was eighth, ahead of Fernando Alonso, who was frustrated by his team's strategy costing him places in the topsy-turvy early part of the race, then pitted a couple of laps too early for slick tyres as the track dried in the closing stages.
That dropped him to last, but he recovered to take ninth, ahead of Mercedes' George Russell, who also stopped early for slicks for the final time.