Norris on pole in Singapore with championship leader Verstappen alongside
Seven-time world champion Lewis Hamilton qualified third with his Mercedes team mate George Russell fourth and McLaren's Oscar Piastri fifth in a top 10 shootout briefly halted by last year's winner Carlos Sainz crashing his Ferrari.
The pole was Norris's fifth of the season, sixth of his career and third in the last four races.
"The car’s been feeling good and when you have a good feeling car and you’re confident, you can go out and push and get the lap time," said Norris, who was fastest in practice on Friday and Saturday.
"So I did the job I had to do today and excited to see what we can do tomorrow."
Verstappen leads Norris by 59 points with seven rounds remaining and the start will be crucial, with overtaking tricky on the Marina Bay circuit.
The final phase of qualifying was halted, with only Piastri and Haas' Nico Hulkenberg having set times, when Sainz slammed backwards into the barriers at the final corner as he prepared to start his flying lap.
"I don't know if it was cold tyres, dirty air or I was... big snap," said the Spaniard over the team radio.
The crash meant Verstappen had his flying lap deleted under yellow flag rules and left seven drivers without a time on the board and only eight minutes remaining in which to set one.
The clock then ticked below four minutes before the two McLarens made the first move.
Norris went top with a lap of one minute 29.525 seconds, with Hamilton second before Verstappen went faster but still 0.203 off the McLaren's pace.
"I'm happy to be on the front row if you look at where we came from yesterday," said Verstappen who was only 15th in second practice and has yet to win in Singapore.
"I'll take second. I'm happy with that...in Singapore a lot of things can happen, but at least we have a shot at it like this."
Hulkenberg starts sixth with Fernando Alonso seventh for Aston Martin and Yuki Tsunoda eighth for RB.
Ferrari's Charles Leclerc had his lap deleted and dropped from seventh to ninth, with Sainz 10th - a dismal outcome for the team McLaren had seen as their main rivals coming into the weekend.
Verstappen's team mate Sergio Perez qualified 13th, the same position he had on the grid in Singapore last year before finishing eighth, on a disappointing night for a driver whose best results have been on street circuits.
Williams drivers Alex Albon and Franco Colapinto will start 11th and 12th.
Daniel Ricciardo failed to make it beyond the first phase of what could prove to be the last qualifying session of his career.
The Australian, only 16th fastest, looks set to be replaced at RB by New Zealander Liam Lawson from the next round in Austin, Texas, as Red Bull assess their options for 2025.