Red Bull’s Sergio Perez in sixth split the Ferraris of Carlos Sainz and Charles Leclerc, who took seventh despite a crash on his first lap that required a pit stop to replace a damaged front wing and left him with bent steering.
McLaren’s Oscar Piastri was eighth ahead of the Sauber of Valtteri Bottas and, delighting his home fans, Zhou Guanyu.
Norris said: “I got a bit nervous. They were conditions where you have to risk a lot and push. I was quick but I kept catching the Ferrari and had to keep backing off.
“I got a good final lap and it was enough for pole. Sad it’s not a proper qualifying but good enough.
“To end up on top is exactly what we wanted and it’s a nice surprise.
“Good day, tricky conditions, always nice to do it like this. Tough, especially with the rain in Q3. Conditions I’ve always loved and always done relatively well in.”
The sprint race will take place at 04:00 UK time on Saturday in a tweak to the scheduling of these weekends for this season.
Norris added: “The pace is good whether it’s wet or dry. The car is feeling good and so am I, and it’s paying off.
“Even in the dry we were pretty good. I don’t think we maybe have the pace of the Red Bull in outright conditions, especially in the dry, but in the wet surprisingly very quick. I could get good tyre temps and push hard.”
Hamilton’s lap was a pleasant surprise for Mercedes, who were struggling for pace in dry conditions.
The seven-time champion managed to just sneak through into the top 10 shootout in ninth place before the rain started and team-mate George Russell failed to do so in 11th place.
“Tricky conditions,” Hamilton said. “Not a lot of grip as you saw for everyone.
“But so happy. As soon as I saw the rain coming, I was getting excited because in the dry conditions we’re not quick enough, so when the rain came I thought I would have a better opportunity and that’s when it all came alive.”
The conditions made for a helter-skelter session with drivers clearly struggling for grip on a track where F1 has not held a race for five years.
Verstappen would have been expected to be on pole – the Red Bull is the fastest car and he is usually so impressive in the wet.
But he mad errors on his first two laps, which led to his times being deleted, and he had to string one together just as the session was ending.
Even so, 2.088secs off pole position is unfamiliar territory for the man who has dominated F1 for the past two years.
Starting fourth, though, Verstappen may still be tough to beat over 19 laps in the sprint.
“It was incredibly slippery,” the world champion said. “I struggled a lot to get the temperature in the tyres. That’s why it was just very difficult to keep the car on track and it never really switched on for me.
“It was like driving on ice and that’s why it’s deserved, where we are in qualifying. It was not really working for me in the wet even though in the dry it looked quite good.”
