Russell had felt on arriving at the track on Saturday morning that he had a good chance of setting pole, and his confidence was well placed.
Like Verstappen using the medium tyre for his final lap, Russell said it was "one of the most exhilarating laps of my life".
He added: "I swear, every corner I was looking at my lap-time delta on my steering wheel and I was going 0.1secs quicker, 0.1secs quicker. I got to the last corner and I was 0.6secs up and I thought: 'This lap's mighty.' I am so pleased with pole."
Verstappen said: "I felt quite good all weekend. The car was in a good window. This track is quite cool. It feels like a big go-kart on the kerbs.
"You need to be quite efficient on the straights, which I think we are. The tough choice was the tyres and I think we chose the right ones. We will see what we can do tomorrow. I am already happy to be on the front row and I hope we can have a solid race."
McLaren had had a difficult weekend and Piastri said he had been "pretty happy with third".
The grid gives the Australian a good chance to extend his championship lead over Norris, who had a tricky session.
He missed the final chicane on his first lap of the top 10 shootout and had to be reminded not to push too much in the braking zones.
His second attempt was slower than Alonso's first and Norris failed to improve on his final run, and was bumped further down by Russell, Antonelli and Hamilton.