Joelinton marked his 100th Premier League appearance with two goals as Newcastle moved up to ninth and sent Norwich closer to relegation.
Kieran Dowell skied two good chances with first-time efforts for the hosts, who are now eight points adrift with five games to go.
Joelinton scored the opener when he swept a lovely finish into the top corner from Allan Saint-Maximin's pass.
He added a second when he slotted in at a tight angle from Jacob Murphy's ball after Newcastle got in behind Norwich's defence.
Those were only his third and fourth goals of the season.
Another Brazilian in Bruno Guimaraes scored United's third, chipping Tim Krul after the Norwich keeper had passed the ball directly to him.
Only Liverpool have picked up more Premier League points in 2022 than Newcastle's 32.
This is the first time since May 2014 that Newcastle have been as high as ninth this far into a season - and that is despite them spending 150 days in the relegation zone.
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