The McLaren pair pushed runaway series leader Max Verstappen of Red Bull down to third for a race in the evening that could see the Dutchman secure his third consecutive drivers' title by finishing sixth or better.
It was the first front row lockout for McLaren since the 2012 Brazilian Grand Prix when the top positions were occupied by Lewis Hamilton and Jenson Button.
"I'm feeling very happy," said Piastri, a 22-year-old Australian rookie. "I might give the FIA five minutes to make sure I am actually on pole, but it was a good lap. I saw Lando on the big screen at the last corner make a mistake,so I'm happy to be here.
"I struggled a bit earlier, but I got it together in the last part of qualifying so I'm happy with that. Max is third so he's not a million miles away so we'll try out best and see what we can pull off."
He later added: "It would have been even better if it had been yesterday," referring to Friday's qualifying session for Sunday's Grand Prix race when he was relegated from the front of the grid after having a lap deleted.
George Russell took fourth for Mercedes ahead of the Ferraris of Carlos Sainz and Charles Leclerc, Nico Hulkenberg of Haas and Sergio Perez in the second Red Bull.
Two-time champion Fernando Alonso was ninth for Aston Martin ahead of Esteban Ocon of Alpine and his team-mate Pierre Gasly with seven-time champion Hamilton qualifying 12th.
Several drivers had their best laps deleted for track limits infringements while several others were reported for running too slowly on non-push laps.
Saturday's qualifying session, known as the sprint shootout, was delayed to accommodate a 'warm-up session' for teams and drivers to become accustomed to changed track limits at Turns 12 and 13.
The changes followed concerns by tyre supplier Pirelli at the damage to their tyres caused by kerbs on Friday.
Circuit officials changed the limits and re-painted kerbs, causing a brief delay before the action began.
The sprint qualifying session began in sweltering conditions with an air temperature of 36 degrees Celsius and the track at 44.
Lance Stroll of Aston Martin was again quickly eliminated, failing to progress from Q1 for a fifth consecutive session. He exited along with Alex Albon of Williams, Yuki Tsunoda of Alpha Tauri, Haas's Kevin Magnussen and Logan Sargeant in the second Williams.
All five were among those to have their best laps deleted.
In Q2, seven-time champion Lewis Hamilton missed the cut.
He wound up 12th behind Pierre Gasly of Alpine, Valtteri Bottas of Alfa Romeo, Liam Lawson of Alpha Tauri and Zhou Guanyu in the second Alfa Romeo.
The final part of the 'shootout' began with Verstappen clocking a fast lap that was promptly deleted for exceeding track limits at Turn Five.
That gave an opening to Sainz and then Perez before Piastri and Norris then took over for McLaren while Leclerc had two laps deleted before Verstappen responded, but only to take third.
