Carlos Sainz put his Williams on the front row alongside the four-time world champion on a tough afternoon for runaway leaders McLaren, who can clinch the constructors' title on Sunday.
Piastri, last year's winner in Baku, qualified ninth without setting a time in the final phase after his biggest mistake so far in a campaign otherwise remarkably free of errors.
"I think the win is ambitious but let's wait and see," said Piastri. "The car has been quick this weekend and hopefully we can use that to make progress."
Teammate and title rival Lando Norris, 31 points behind, will start only seventh after going breathtakingly close to hitting the wall and missing a big chance to put distance between himself and the Australian.
Racing Bulls' Liam Lawson qualified a surprise third with Mercedes' Kimi Antonelli fourth in an all-rookie second row on a grid with a distinctly different look even if the pole was a second in a row for Verstappen after Monza.
'Very difficult session'
"It was a very difficult session. In the final lap, you just have to send it," said Verstappen after a last effort that was 0.478 quicker than Sainz.
"I wasn’t even on the best tyres that I wanted - because of all the red flags you basically run out of tyres."
Leclerc had been bidding for a fifth successive Baku pole but it all went wrong when the Monegasque speared into the barriers on a track made slippery by rain.
Only three drivers, with Sainz leading Lawson, had set a time at that point of the final phase.
Ferrari's seven-times world champion Lewis Hamilton qualified a disappointing 12th - after the optimism of going fastest on Friday - with Aston Martin's Fernando Alonso 11th.
"I honestly thought I was going to be shooting for pole today, so it’s kind of a bit of a shock," said Hamilton.
First phase stopped three times
The first phase was halted three times, with Alex Albon hitting the wall in his Williams and then Nico Hulkenberg going straight into the barriers at turn four and smashing the front wing and floor of his Sauber.
Argentine rookie Franco Colapinto, his seat for next season in doubt, then crashed his Alpine heavily in the closing seconds -- repairs to the barrier delaying the start of the second phase.
Hulkenberg's crash brought out the red flags with six minutes and 36 seconds remaining, putting pressure on Antonelli as the only driver still without a time.
Both Alpines as well as Haas's Esteban Ocon joined Hulkenberg and Albon in failing to make it through the first phase.
Haas's Oliver Bearman halted the second phase, the fourth stoppage of the session, when he hit the wall with just under 12 minutes remaining.
"Sorry guys, so stupid," said the Briton over the radio after stopping on track and informing the team that the car was 'broken'.
Leclerc's crash was the fifth stoppage and Piastri then triggered the red flags again with three minutes and 41 seconds remaining.