Mercedes' Lewis Hamilton completed the podium at the Hungaroring where McLaren locked out the front row of the grid for the first time since 2012.
"Amazing day for us as a team. Oscar controlled the race and he deserved it today," said Norris, who had been urged by his team to let Piastri through in the closing laps.
"This is really the day I dreamed of as a kid. It was a bit complicated at the end," said Piastri, referring to the repeated demands from the team to Norris to let him through for the win.
Max Verstappen, who still leads the championship from Norris, finished fifth after a tetchy race when he had to give a place back to pole-sitter Norris at the start after going wide as Piastri took the lead at the first corner.
A late lunge by the three-time world champion past Hamilton forced his Red Bull off the track, with the stewards reviewing the incident.
