Norris finished Q3 just 0.241 seconds off Max Verstappen’s pole time, with team-mate Oscar Piastri a further tenth behind him.
He had clinched provisional pole in the closing stages before he was demoted by the reigning world champion’s best time, a 1m26.720s.
It marks the Woking-based team’s best qualifying of the year, with Norris’ previous best a third-place start in Barcelona.
Norris joked that Verstappen had “ruined everything” after he was pipped by the Dutchman’s final attempt, but said he was looking forward to Sunday’s race.
McLaren’s last British GP win came in 2008, courtesy of Lewis Hamilton.
Asked just how close he was to pole, Norris said: “I was close. I was P1 in Q1. I think it was P1 in Q2. And close, two tenths to P1 in Q3, so pretty insane.
“And my last lap was a good lap. I could hear Zak on the radio on the in lap, that was the best thing ever.
“Great for myself but also for both of us to be second and third was pretty amazing for the whole team.”
He added: “It's always Max, he always ruins everything for everyone. I am very happy. I've got a big thanks to the crowd here, it's been a very special day for us. I look forward to tomorrow.”
Piastri said the MCL60 was “a rocket ship” in Q2 and Q3 and Saturday marked a “massive result” for the team, securing him his best ever F1 qualifying result.
Speaking after qualifying, Piastri said: “What a qualifying session. We were almost out in Q1, and then yeah, car was a rocket ship in Q2 and Q3. Feels like I put a pretty good lap together.
“So yeah, massive result for the team as well. To be consistently have both of us up here is a mega result. Now we've just got to try and stay there for tomorrow.”
Piastri said the team had “a bit of a rough day” on Friday, with Norris eighth and Piastri 10th in FP1 before Piastri finished ninth and Norris 14th in FP2.
The Australian added: “I think we were a bit quicker than what we looked, just struggling to get the car in the right window and I think we did a better job of that today.
“I think these conditions as well. We do really well in them all the time. Barcelona was a similar story Monaco, Canada as well.
“So we've had a fair few moments like that. But yeah, to pull that off in Q3 like that was a mega result and obviously both of us have now got the new bits on our car and they're working very well.
“So yeah, I can't thank the team enough for continuing to push from where we were at the start of the season to where we are now. It's a massive step in the right direction.”
Verstappen took his sixth pole of the season on Saturday in a “crazy qualifying” and said he was “quite surprised” to see the McLaren pair behind him.
He said: “It's been quite hectic and also quite slippery in some places. But yeah, we did our laps, and then Q3.
“I was quite surprised to see them two there, but it's great for McLaren as well to be here. And from our side very happy to be on pole.”
