Using a safety car period, the 24-year-old Bristolian managed to overtake Verstappen in his 110th Formula One race.
He crossed the finish line seconds ahead of Verstappen, who finished in second position, and Charles Leclerc of Ferrari, whose teammate Carlos Sainz finished in fourth.
Sixth-place finisher Lewis Hamilton achieved his best finish of the year.
“Whooooooooooo, whooooooooo, I love you all,” said Norris over the announcer.
“Many thanks for it. Will Joseph, Norris’s racing engineer, said, “We did it.” We succeeded.
“I suppose that’s how things are done. Lastly. Oh, I’m overjoyed.
I suppose that’s the process. Lastly. Oh, I’m overjoyed. When I arrived this morning, I was aware of it. I mentioned that there are a lot of opportunities now. I’m so glad you and I both got it right.