After taking the lead at the start, when he passed pole-sitting teammate Lando Norris, the Australian pulled clear to control the race until losing out in the pit stops, when the Briton gained an advantage.
McLaren pitted him first for an 'undercut' to fend off seven-time champion Lewis Hamilton of Mercedes who finished third after surviving a skirmish with Red Bull's Max Verstappen.
This resulted in Norris, who had superior pace, inheriting the lead, leading into the closing laps and receiving repeated orders to give back his position - and victory - to Piastri, who was struggling to close the gap.
After prolonged strained dialogue on the team radio, Norris agreed to slow down and let Piastri pass him with three laps remaining, despite knowing he was losing a chance to cut substantially into Max Verstappen's lead in the drivers' championship.
"A very special day for me," said Piastri.
"This is the day I've dreamed of as a kid, to be standing on the top step of an F1 podium.
"A bit complicated at the end, but I put myself in the right position at the start."
He added he was unfazed by the team orders arguments played out over the closing laps as McLaren asked Norris to slow down, preserve his tyres and allow Piastri to catch and pass him.
"The longer you leave it, of course, you get a bit nervous, but it was well-executed by the team and it was the right thing to do. I had put myself in the right position at the start," he said.
"With the different strategies we had, my pace probably wasn't as quick as I would have liked in the last stint, but I was still in the right position to make it happen."
'Tough' for Norris
Piastri also revealed he and Norris had discussed their rules of engagement in relation to the race in detail late on Saturday and early on Sunday.
"So long as we finished one-two and did the right thing it was fair," he said.
"We had a lot of good open discussions and with a lot of respect and trust in each other."
Norris admitted it was emotionally difficult to give up a victory.
"Tough, tough for anyone," he said.
"They made me box first to give me the chance to lead and pull away, but I know it was fair to give it back and Oscar deserved it.
"He has done a lot for me and I don't want to be unfair. He should have had it in the first place and for me, it was down to my bad start."
"It's been an amazing day for us. That's the main thing. It's been a long journey to achieve this on merit and that’s exactly what we did. We are a long way ahead of the rest and we did it in style."
Hamilton finished third for Mercedes after surviving a late collision with Red Bull's three-time champion and series leader Verstappen, who flew off but recovered to finish fifth.
Charles Leclerc came home fourth and Ferrari teammate Carlos Sainz sixth, sandwiching a grumpy Verstappen who was called to see the stewards to explain his collision with Hamilton.
