Starting from third following Lewis Hamilton's three-place grid drop for a yellow flag infringement in qualifying, Norris dropped to fifth in the opening laps.
Late five-second penalties for Hamilton for contact with Alexander Albon, and for Sergio Perez for pitlane speeding, Norris found himself in podium contention in the closing laps.
An aggressive move on the Racing Point at Turn 3 gave him some clean air, which he used to post the fastest lap of the race on the final lap and end up within 4.8s of Hamilton at the chequered flag – netting him a first F1 podium in third by just two tenths on corrected time.
Commenting on the race, Norris said: "I'm speechless. I think there's a few points in the race where I thought I kind of fudged it up quite a bit.
"I dropped to fifth with a few laps to go, Carlos [Sainz] was almost going to get past me.
"But I didn't give up and I managed to get past Perez and ended up on the podium.
"I mean it's a long race, but I kept going and kept giving my all, [I had] a pretty cool last few laps trying to push as much as I can. I am a bit out of breath, but I am so happy and proud of the team.
"Considering where we were a few years ago to now, I think it's a pretty good achievement, so I'm proud to be part of it."
Norris says he was aware of Perez's penalty, but not Hamilton's at the time, and his hard pass on the Racing Point was more down to being vulnerable to attack from behind owing to struggles following cars.
"The last few laps, when I had to get past Checo, I knew he had a five second penalty but nothing more than that," he added.
"I always seemed to struggle when I was close to the cars ahead and more vulnerable to the guys behind.
"So I knew - not just because of Lewis, because I didn't know at the time [he had a penalty], I knew I had to try to get past him.
"And then when the Lewis penalty came, I knew I had to turn it up a little bit."
McLaren had to lay off around 1200 of its staff across its company during the coronavirus-forced lockdown having been hit hard by the financial fallout wrought by the pandemic.
All of that came months after a member of its race team tested positive for COVID-19 on the eve of the Australian Grand Prix, which led to its withdrawal from the race and ultimately the outright cancellation of the event.
Team boss Andreas Seidl noted that there was "nothing better" than Norris' result for the team after such a "difficult period".
"Obviously I'm just proud to be honest to be part of the team today," Seidl told Sky Sports F1.
"There could be nothing better for the entire team, especially after this difficult period, and such a result to get P3 and P5 I think, it's simply unbelievable for the entire team."