The McLaren driver headed Max Verstappen and Charles Leclerc to the chequered flag to take his first F1 win on his 110th race start.
Norris benefitted from a mid-race Safety Car that afforded him a cheap pit stop, before strong pace allowed him to ease clear out front in the latter half of the race.
“To get that first victory is always incredible,” Norris said.
“And I’ve, of course, had my moments where we’ve been close, and I’ve never been able to convert it into the win. But I wasn’t worried.
“As much as a lot of people doubted that I was going to be able to put it together and win a race, I wasn’t worried.
“I’ve kind of been more confident than ever this year that I’ve got what it takes and the team have got what it takes and I was patient with it,” he added.
“I’ve just been doing my job and executing my races, executing my qualis, and doing what I can do best.
“I knew my time was coming.
“There was kind of that spark, and we maybe lost it a little bit into Saturday, but today it definitely came back and turned into a little fire.
“It was an incredible race.”
Having originally been scheduled to fly out of Miami on Sunday evening, McLaren boss Zak Brown changed those plans so the win could be celebrated.
“I was meant to fly, but Zak just told me we fly tomorrow,” Norris noted.
“I’m not going to sleep. I’m going all the way. I might have more than just a scratch on my nose tomorrow, so I don’t care.
“But I’ll enjoy my time. This only happens once when you’re taking your first win. It’s nice to do it here in Miami.
“I was kind of hoping it was going to be at Silverstone, but I’ll try to do it there as well.”
Verstappen, who won the first two editions of the Miami Grand Prix, added: “It’s definitely better here, mate, to go out!”
It left Norris to declare, “Tonight’s going to be a good night.”
Norris’ McLaren team-mate, Oscar Piastri, finished 13th after the Safety Car emerged at an inopportune time for the Australian.
He’d run second through the opening stint of the race and looked Verstappen’s nearest challenger before Kevin Magnussen punted Logan Sargeant into the fence.
After the restart he didn’t have the same pace he’d enjoyed earlier and eventually fell victim to Carlos Sainz, who tagged the McLaren’s front wing as he passed on Lap 39.
That essentially ended Piastri’s race as he was forced to pit for a replacement, the Ferrari driver being pinged with a five-second penalty for the incident.
For Norris, the win leaves him tied with Sainz in fourth in the drivers’ championship on 83 points.
McLaren now sits in a comfortable third in the constructors’ title fight, with 124 points – almost double Mercedes in fourth and 63 back from Ferrari in second.
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