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Norris was fourth at the flag, inheriting two places on the penultimate lap when Sergio Perez and Carlos Sainz crashed exiting Turn 2.
Together with the bonus point for fastest lap, the McLaren driver banked a total of 13 points in the championship.
His title rival, Max Verstappen, started the race sixth and likewise gained two places in the final laps, trailing Norris to the flag in fifth.
That brought with it a further 10 points in his championship quest, his advantage at the top of the standings reducing by three points to 59.
But while he conceded ground, the result can still be regarded as a positive one for Verstappen, a point the Red Bull Racing driver made.
“Probably positive for me,” he said of the race.
“Of course, I would have liked to extend the gap but with our race today, I’m still happy that it was only that.”
For Norris, it could have been far worse.
Having blamed a yellow flag for his Qualifying 1 exit in the all important session, he climbed back through the pack well, achieving a result well above the team’s expectations.
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“We would have been happy with eighth,” Norris confessed.
“We just expected the top four teams to go, obviously me being the eighth car so we expected them to go and beat everyone else.
“But I got into 10th quite quickly. I was in 10th already I think five, six laps into the race.
“That’s a good kick start to the whole day.”
He then found himself caught behind Alex Albon, remaining there until the Williams pitted to release the McLaren.
“As soon as Alex boxed, my pace was I think the best on track, even on the hards from the beginning of the race.
“Eight was our target, but we know doing a hard start, and if things went well, better was possible but it’s hard to anticipate and know what that was going to be.”
But while Norris limited the damage, just as Verstappen did, his chances of winning the drivers’ championship have decreased.
Heading into the event, the 17th round of the season, Norris trailed Verstappen by 62 points.
That translated to the need to out-score Verstappen, on average, by 7.75 points each weekend to the end of the season.
By only closing in three, that figure has risen to 8.43 points per event.
While that in itself is comparatively minimal, in reality it ramps up the degree of difficulty; Norris essentially needs to finish no worse than fifth (or sixth with a bonus point) even if Verstappen fails to score simply to remain on target.
Should the Dutchman be eighth or higher, Norris must win the race in order to erode Verstappen’s advantage in any meaningful way.
While still possible, the degree of difficulty has increased in the wake of the Azerbaijan Grand Prix.
