Joelinton marked his 100th Premier League appearance with two goals as Newcastle moved into the top half 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.