Oscar Piastri was fourth, meaning his lead over McLaren teammate Norris at the top of the drivers' standings was cut to 22 points.
The 27 points scored by Piastri and Norris were more than enough for McLaren to equal the record set by Red Bull in 2023 by winning the team title with six races to spare.
It was McLaren's second title in a row and 10th in the team's history, and was won despite their two drivers clashing in the opening turns with sparks flying as Norris got the better of Piastri.
"They have driven brilliantly all season. You can't win the constructors' without two awesome racing drivers," said McLaren CEO Zak Brown.
"As you can see, we are letting them race. That was a bit more of a nail-biter there, but they race hard, they race clean, they race to win.
"A lot of racing to go and hopefully a lot more victories for both of them."
Russell was overjoyed to win the sweltering night race for the first time, saying it was payback for his final-lap crash two years ago while pushing for victory on the Marina Bay street circuit.
"It feels amazing, especially after what happened a couple of years ago. That was a bit of a missed opportunity, but we more than made up for it today," said the Englishman, who started on pole.
"We don't really know where this performance came from, but really, really happy."
Dutchman Verstappen was relieved to finish second for Red Bull.
"I think the whole race was quite difficult, more difficult than I hoped for, for a lot of different reasons," said the four-time world champion, who remains third in the title race, 63 points behind Australia's Piastri.
"There's a few things that we need to understand why they went wrong today.
"But around here even if you have more pace, you can't pass without anything crazy happening. So I think second was the maximum result."
Not fair
Kimi Antonelli crossed the line fifth in a Mercedes, followed by the Ferraris of Charles Leclerc and Lewis Hamilton.
Fernando Alonso, Oliver Bearman and Carlos Sainz rounded out the top 10.
Russell got away cleanly from Verstappen at the start, but the drama all happened behind.
Norris, from fifth on the grid, darted past Antonelli and dived up the inside of Piastri, who started third, at turn one.
Norris was flying and clipped the back of Verstappen on the way to barging his way into third place in a wheel-to-wheel clash with teammate Piastri.
The championship leader was not impressed. "So are we cool with Lando just barging me out of the way there? What's the go there?" Piastri complained on team radio.
His anger was not appeased by the team saying they would discuss the incident after the race and not order Norris to swap places.
"That's not fair. I'm sorry, that's not fair," Piastri said.
Norris disagreed. "It was good racing," he said.
After pitting for hard tyres, the leading four emerged in the same order, though Verstappen was reporting downshift problems with his gearbox that felt "like a handbrake".
By lap 41 of the 62 Norris had the ailing Verstappen in his sights but found it impossible to get close enough to pass on a circuit where overtaking is notoriously tricky.
"Max didn't make any mistakes. I gave it my all today and got close," said Norris.
The night race in tropical Singapore was declared the first official Formula One "Heat Hazard".
That means all drivers had to have liquid-cooled vests available, though wearing them was not mandatory.
However, with the air temperature around 28C for the race, rather than the expected 31C, some opted not to use the new equipment, including Verstappen.
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