Norris needs to close on Verstappen by just over eight points a race on average to beat the Dutchman to the championship – more than the gap between first and second places.
Norris’ team-mate Oscar Piastri, winner of the Azerbaijan Grand Prix last weekend, could manage only fifth.
And it was a poor session for Ferrari, for whom Sainz won in Singapore last year.
In addition to the Spaniard’s crash, team-mate Charles Leclerc had his lap time deleted for exceeding track limits and will start ninth.
Leclerc had in any case only managed to set seventh fastest time, slower than the Haas of Nico Hulkenberg, who took an impressive sixth place ahead of Aston Martin’s Fernando Alonso and RB’s Yuki Tsunoda.
Norris said: “I was finding it tough to progress much and the guys around me were getting quicker and quicker putting me under pressure. But it was good enough for pole. I felt confident all weekend. Maybe not so much in qualifying, but we got the job done.”
Verstappen, who had been unhappy with his car in Friday practice, said: “The whole of qualifying went quite well. We managed to improve the car.
“I am happy to be on the front row if you look at where we came from yesterday. Everyone only has one lap so you don’t want to overdo it. I take second, I’m happy with that.”
Hamilton’s third place is his best qualifying since he was second on the grid at the British Grand Prix.
He said: “Qualifying has been a disaster for me all year long and I have just been working and working and working trying to get myself back up there and all of a sudden the car came to me for the first time in a long time in qualifying.
“We have been moving up and down on balance. We have changed everything and the mechanics have been faultless and I hope we are in a good position to fight for the front tomorrow.”
Hamilton edged out Russell by just 0.026secs, while Piastri ended up 0.428secs slower than Norris.
The Australian has promised to help Norris in his title fight with Verstappen if he can, but starting three places behind the Red Bull will make that difficult.
Ferrari’s poor pace, meanwhile, seemed to be caused by a struggle with tyres and getting the fronts and rears at the right temperature at the same time.
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