Verstappen had taken provisional pole with a time that looked hard to beat, and went faster with his final effort, but Norris took it to another level to end the session 0.020 quicker than his Dutch rival.
The pole was only the second of the Briton's Formula 1 career, the last coming at the Russian Grand Prix in 2021.
"It was pretty much a perfect lap. You know when you are on a good lap and you are getting excited," Norris said after setting a time of one minute 11.383 seconds.
"It has been close all weekend but really it was about a perfect lap and that is what I did today."
Championship leader Verstappen had set the standard with a lap of 1:11.673 as his first effort and improved the time on his second run to an impressive 1:11.403.
"I was quite happy in qualifying. I got a nice (aerodynamic) tow from Checo (team mate Sergio Perez) in Q3 (the final phase) but unfortunately it was not enough," he said.
"That's how it goes sometimes, but overall we can still be happy with that performance and it's all to play for tomorrow."
Norris was the fourth different driver on pole in the last four races, another sign of the battle between the top teams getting ever closer and Red Bull's dominance being steadily eroded.
Fire drama
The pole was also a boost for McLaren whose day started with unwanted drama when
their paddock hospitality unit caught fire, with one team member taken to hospital as a precautionary measure.
Norris was one of those in the structure when it was hurriedly evacuated, with the driver losing a pair of shoes and also having to prepare for qualifying without his usual music.
"I have one or two sets of everything and they managed to get some stuff out, some of it smells pretty bad from the fire," he said.
Mercedes's Lewis Hamilton was third fastest by a wafer-thin margin of 0.002 over team mate George Russell.
"Slowly the car is crafting into a racing machine with which we can hopefully fight the guys in front," said seven-times world champion Hamilton, who outqualified his team mate for only the second time in 10 races this season.
Russell had angrily questioned Hamilton slowing ahead of him to prepare for his flying lap but accepted later that he had been "a bit hotheaded" on the radio.
"As a team, we are in the mix. I think Lando did a really great lap. I don’t think Lewis or I put it together perfectly. I felt there were two tenths on the table," he said.
"We’re on the second row and I think it’s going to be a good fight with Lando and Max tomorrow. I think we’ll probably just have the edge on Ferrari."
Ferrari's Charles Leclerc and Carlos Sainz, preparing for a home race, filled the third row.
Pierre Gasly boosted struggling Alpine's morale with seventh place on the grid for the Renault-owned team and Perez was eighth but will drop three places due to a grid penalty.
That will lift Alpine's Esteban Ocon to eighth with McLaren's Oscar Piastri ninth, after having his final laps deleted for exceeding track limits, and Aston Martin's home hero Fernando Alonso 10th.