George Russell qualified second fastest for Mercedes but will drop down the grid due to a five-place penalty, with newly-crowned champion teammate Kimi Antonelli replacing him on the front row.
Ferrari's Charles Leclerc was fourth fastest but has a 10-place grid drop, which meant Racing Bulls' Liam Lawson will start third and Audi's Nico Hulkenberg fourth.
"Today is about getting on pole. It doesn’t really matter how you do it, so very pleased with that,” said Norris after a qualifying that started and was then halted for nearly an hour and a half.
With the rain getting heavier and leaving no chance for anyone to improve on their times in the final phase, the clock ticked down for the last five minutes without a car on track.
Russell was furious not to be fastest, wanting to make a statement despite knowing he had a penalty but having to abort his quick lap when he came up behind Williams' Alex Albon.
"Mate, you told me to back off! What the hell?," he shouted over team radio after realising his day was done. "P2. I don't want to be P2. I want to be P1."
Isack Hadjar will line up fifth for Red Bull, with Oliver Bearman sixth for Haas. Russell will start seventh and Fernando Alonso judged the conditions well to finish eighth as Aston Martin reached the final phase for the first time this season.
Albon qualified ninth and McLaren's Oscar Piastri 10th.
Red Bull's four-time champion Max Verstappen, with a 30-place grid penalty due to power unit changes, sat out the second phase that was red-flagged when Haas debutant Rafael Camara crashed.
Ferrari's Lewis Hamilton, who will also be starting at the back of the grid due to engine penalties amounting to an impossible 65-place drop, also went out.
Conditions had eased in Q2, with double world champion Alonso leaning on his experience and the first to switch to intermediates.
Camara earlier had the home crowd roaring when he reached the second phase but fellow-Brazilian Gabriel Bortoleto had a nightmare for the second year running when he went off and qualified last.
Bortoleto slid into the tyre wall with seven minutes to go but returned to the track and made his way back to the pits, where the damage proved too extensive to fix in the time available.
Both Alpines also went out at the first hurdle, along with the two Cadillacs and Aston Martin's Lance Stroll.