The pole on a hot and humid evening was the Spaniard's second in succession, as well as the Italian team's third in four races.
"I knew I had the pace. I knew I could do it," said Sainz, who was also on pole at Ferrari's home Italian Grand Prix two weeks ago but finished third behind the Red Bulls.
"Now more than ever it’s true that I want to get it done, not like in Monza," said Sainz, who has Mercedes' George Russell lurking alongside on the front row.
"Tomorrow we will try and get that win."
Red Bull's runaway championship leader Max Verstappen and teammate Sergio Perez both failed to make the top 10 shootout, an astonishing turnaround for a team that had been in a league of their own but suddenly looked lost.
Suddenly Red Bull were nowhere, Verstappen's hopes of a record-extending 11th successive victory seemingly gone on a city circuit where overtaking has never been easy.
The Dutch 25-year-old qualified 11th and then faced three post-qualifying stewards' enquiries for allegedly impeding, two of which drew reprimands and the other no further action.
"That was...shocking, absolutely shocking experience," Verstappen said over the radio.
"I knew it was always going to be tough to put it on pole but this I didn't expect," he told Sky Sports television, saying the car had been really difficult to drive.
Asked if victory was now off the table, the Dutchman, who has won 12 races this season and is yet to finish lower than second, told reporters: "Yeah, you can forget about that.
"You can't pass. On other tracks you can start last and win the race. But not here."
Perez, last year's winner, spun and qualified 13th for what will be his 250th Formula One start.
ALL TO PLAY FOR
Without the Red Bulls on track, rivals had everything to play for and Sainz made his lap count with a best time of one minute 30.984 seconds.
Last year's pole sitter Leclerc looked like securing a front row sweep for the Italian team but Russell came through with a time just 0.072 slower than Sainz.
McLaren's Lando Norris will start fourth with Mercedes' Lewis Hamilton fifth on the grid and Haas's Kevin Magnussen sixth.
"I really hope George gets a great start tomorrow and puts some pressure on those Ferraris. It would be great for him to get a win," said Hamilton.
Aston Martin's Fernando Alonso starts seventh with Alpine's Esteban Ocon eighth, Haas's Nico Hulkenberg ninth and AlphaTauri's Liam Lawson 10th.
Aston Martin's Lance Stroll crashed heavily at the final corner in the dying seconds of the opening session, bringing out the red flags and delaying the start of the second phase.
Stroll clambered out of the wrecked car, which smashed into the barriers almost head-on and so hard that the front left wheel was ripped off, and was taken to the medical centre for checks.
The red flags caught out McLaren's Australian rookie Oscar Piastri, whose final flying lap had to be aborted and left him 17th on the grid at a tight track where overtaking is a challenge.
"It should have been enough to get through," he said of his lap up to that point. "It makes our evening tomorrow very difficult."
