In an example of perfect teamwork, Norris led from pole to flag before slowing to allow Piastri to win by 0.136 seconds ahead of Mercedes' George Russell in third.
The result secured 15 points for McLaren in their bid to resist Ferrari, extending their advantage to 34 points with two Grands Prix remaining.
Ferrari's Carlos Sainz finished fourth ahead of team-mate Charles Leclerc, seven-time champion Lewis Hamilton in the second Mercedes and Haas' Nico Hulkenberg, Max Verstappen, Pierre Gasly of Alpine and Kevin Magnussen in the second Haas.
"It was defence all the way the whole race for me," said Piastri. "I had a good start and then Turn One, but not the pace I needed and then I was struggling, but it was great teamwork and a great McLaren 1-2!"
Norris said he had disobeyed the team to give Piastri his victory as his payback for the Australian's assistance at the Brazilian Grand Prix sprint when he was hoping to keep alive his bid for the drivers' title and Piastri gave him the victory.
"We scored the 1-2 we were aiming for and we have maximum points so I am happy for the team… Perfectly-executed! We did our job. It ended a bit closer than I wanted, but I had this planned since Brazil and it's what I wanted. The team told me not to do it, but I did it the best way I could."
Russell said: "We were very close in T1 a couple of times, but Lando backing up to help Oscar with DRS… It was pretty infuriating to be honest, but I understand why."
'Thanks for the team-work'
The race was preceded by much paddock discussion of Hamilton's speed, or apparent lack of it in Friday's qualifying, the Briton claiming he was "definitely not fast anymore" after being outqualified again by Russell.
His former McLaren team-mate Jenson Button, the 2009 world champion, reacted by saying: "it's getting to him a little bit… But, that's the thing with Lewis.
"You expect him to get more in qualifying. It hasn't happened, but we know his pace and he's still quick. It is in there."
As if to prove it, Hamilton made a blistering start by overtaking Leclerc as Norris led from pole, with Piastri passing Russell for second. Verstappen, struggling with oversteer, dropped to eighth.
"I'm trying to five Oscar DRS every lap," said Norris, in hope of helping McLaren claim a valuable one-two.
By lap 10, Norris was ahead of Piastri by three-tenths with Russell nine-tenths back in third, ahead of Sainz, Hamilton, Leclerc and Hulkenberg. Most were in a 'DRS train' and preserving their tyres for the final laps.
On lap 13, a masterful move by Leclerc into Turn One saw the Monegasque fight his way past Hamilton, the pair running wheel to wheel before the Ferrari pulled away.
Russell was also busy hunting down Piastri and failed in a bid to pass on lap 14 while Norris reported his front tyres were losing performance -– yet he clocked a fastest lap to stay on top, hoping to drag his team-mate with him.
The order remained unchanged going into lap 18, the penultimate, with Russell attacking again at Turn One on lap 19, Piastri defending adroitly to stay second until they reached the chequered flag where Norris backed off to swap positions.
"Thank you for the team-work, much appreciated," said the victorious Piastri as he crossed the line to the relief of team boss Zak Brown. "It's what we wanted."
