Charles Leclerc puts Ferrari on pole in Baku for fourth year in a row
McLaren's Oscar Piastri joined Leclerc on the front row with Ferrari's Carlos Sainz and Red Bull's Sergio Perez together on the second row.
Red Bull's world championship leader Max Verstappen, 62 points ahead of Norris with eight rounds remaining, qualified only sixth fastest with Mercedes' George Russell ahead of him in fifth.
The pole was the 26th of Leclerc's career and third of the season, but he has yet to convert the top slot into victory in Baku where champions Red Bull have won for the past three years.
"It's one of my favourite tracks of the season, I really like it," said the Monegasque, who won the previous race in Italy but crashed in Friday practice in Baku.
"In that last lap, I went for it a bit more and the lap time came very nice. The car felt really good, everything felt great and it's amazing to be on pole.
"Hopefully we can play a team game tomorrow to win... in the past, we were very strong in qualifying and struggling a bit in the race. This year we have a stronger race car, so I hope that we can finally make it tomorrow."
Leclerc's time of one minute and 41.365 seconds was 0.321 quicker than Piastri's best effort.
The Australian had been third after the first flying laps, with Ferrari one-two, but found some extra speed with his final effort.
"I got a little bit close to the walls on the last lap; just tried to get the most out of it," said Piastri, who agreed last week to obey team orders to help Norris but is unlikely to be in a position to be asked now.
"This track rewards commitment and the last lap in Q3, I knew I had a little less to lose, so just tried to maximise the car."
Norris could only watch as a spectator after going out at the first hurdle when a yellow flag was displayed briefly at the end of the first phase and scuppered his last attempt to get out of the bottom five.
"Everyone did their second laps and I didn't," he told Sky Sports television.
"It was unlucky, that's all. I had to lift.
"There's a long race ahead, we have some good tyres in the bank, I'm still hopeful we can get a good result. I think the car is quick. Bit frustrating, but nothing I can do."
Norris could move up a place after Alpine's Pierre Gasly, who qualified 13th, was summoned by stewards for a fuel flow irregularity.
McLaren boss Andrea Stella said the team were discussing with the governing FIA why the yellow flag was triggered by Esteban Ocon's Alpine, which had not stopped.
"There was a situation that ideally, and I think by the regulations, shouldn't have happened. We paid the price but we don't give up," he added.
Seven-times world champion Lewis Hamilton qualified seventh with Aston Martin's Fernando Alonso eighth and the Williams of Argentine rookie Franco Colapinto and Alex Albon ninth and 10th.
Albon's hopes of moving up were wrecked when he left the pits with a yellow cooling fan lodged in the car's airbox.
He stopped, removed the bulky item from behind his head and threw it onto the side of the track but missed the chance of a final lap.
"I'm not kicking the team. I feel like sometimes these things can happen and we've just got to make sure it doesn't happen again," said the Thai.