Piastri took the place after Antonelli made a mistake and ran wide at the final corner as the race was restarted.
McLaren chose to ask him to let the Mercedes back past rather than argue his case with the stewards.
The early laps of the race provided plenty of entertainment as Russell and Hamilton swapped the lead.
Hamilton moved up from fourth on the grid to slip by Norris around the inside of Turn One, before making an opportunistic and improvisational move down the inside of Turn Nine to grab the lead.
Russell easily passed him back down the straight to the hairpin of Turn 14 before Hamilton attacked and passed him around the outside of Turn One at the start of the second lap.
Russell passed the Ferrari back on the straight on lap three, before Hamilton took back the lead into Turn One on lap four, only for the Mercedes to claim the place for good a lap later.
Hamilton was then left to fend off Leclerc and the two Ferraris battled for three laps before the Monegasque took the place into Turn One on lap eight.
Hamilton fought back to challenge Leclerc into Turn One at the start of the following lap and Leclerc had to defend all the way to Turn Six before consolidating his position.
Antonelli then passed Hamilton into the hairpin on lap 11 before Audi’s Nico Hulkenberg stopped in the run-off area at Turn One with a technical failure to bring out the safety car.
That triggered pit stops for fresh, soft tyres for the leading cars and Hamilton and Piastri lost out a little as they had to stack behind their team-mates, and Antonelli dropped back as he served his penalty.
Piastri was ordered to give fifth place back to Antonelli soon after the restart.
Behind the leading cars, Liam Lawson moved up from ninth to seventh when Racing Bulls decided not to stop under the safety car and he was able to fend off Haas driver Oliver Bearman who took the final point.
Max Verstappen, who had dropped to 15th on the first lap with a slow start, moved up to ninth, on Bearman’s tail at the end, just missing out on a point.