It was the first time this season, at the 11th round, that leaders Mercedes had not secured pole position for a Grand Prix.
Hamilton joined Norris on the provisional front row, subject to a stewards' enquiry, after the seven-time world champion missed out by a mere 0.012 of a second on what would have been his first pole since Hungary three years ago.
"Very happy to be back on top," said Norris, last year's winner in Hungary who now has a strong chance of repeating the feat at a circuit where overtaking can be tricky.
"It's been tight the whole weekend, but we seemed quick from the first laps we did, and we brought a few upgrades.
"I’m in the best position for the start. But these guys (Ferrari) get good starts and have been fast all season."
Ferrari's Charles Leclerc qualified in third place with Mercedes's Formula 1 leader Kimi Antonelli, winner of six of 10 rounds and 45 points clear of closest rival Hamilton, completing Sunday's second row.
Hamilton under investigation for possible impeding
Hamilton, chasing a record-stretching 10th pole in Hungary, was fastest after the first runs but the pole got away at the end and he also faced an investigation for possible impeding.
"My mind is just elsewhere this second because once I finished my lap I don't know if I got in the way of someone. So I'm trying to just run that through my mind because I wasn't aware anyone was coming," he said.
McLaren's Oscar Piastri qualified fifth, alongside Red Bull's four-time world champion Max Verstappen, with the Australian possibly blocked by Hamilton.
"It was just Lewis not looking in his mirrors," Piastri said.
Verstappen spun at the end, forcing Antonelli and teammate George Russell to slow. The Briton qualified seventh for Mercedes, another blow to his chances of closing the gap to Antonelli in the last race before the August break, with Red Bull's Isack Hadjar eighth.
Rookie Arvid Lindblad qualified ninth for Racing Bulls with Nico Hulkenberg 10th for Audi.
Struggling Aston Martin made a minor breakthrough with Fernando Alonso, four days short of his 45th birthday, making the second phase of qualifying for the first time this season in the team's upgraded chassis.
The Spaniard will start 16th with teammate Lance Stroll 20th but at least ahead of both Cadillacs.