Joelinton, on his 100th appearance for the black-and-whites, hit a six-minute double in the first-half and Brazilian compatriot Bruno Guimaraes capitalised on a mistake from Tim Krul and produced an excellent chip to record a fourth top-flight win in a row for the first time since April 2018.
The result sees The Magpies move into ninth on 43 points with four game remaining, while The Canaries stay rock-bottom and are set for an instant return to the Championship.
Story of the game
Eddie Howe hinted in his pre-match press conference that he might "freshen things up" ahead of the trip to East Anglia with The Magpies in action for the third time in the space of six days and decided to make four changes from the 1-0 victory over Crystal Palace.
Fabian Schar, Jonjo Shelvey, Chris Wood and the match winner on Wednesday evening Miguel Almiron all dropped to the bench in place of captain Jamaal Lascelles, Emil Krafth, Jacob Murphy and Joe Willock.
Newcastle had the first sight at goal just seven minutes in as Guimaraes played a dangerous ball over the Norwich defence for Murphy to attack and the winger hit a first-time volley narrowly wide of Krul's right-hand post.
Yet, just two minutes later, in a carbon-copy of Newcastle's opportunity, it was The Canaries who almost broke the deadlock as Mathias Normann picked out Kieran Dowell, who got in behind left-back Matt Targett but diverted his effort over the crossbar.
Norwich came close again and should have scored 17 minutes in when Pierre Lees-Melou weaved through a couple of challenges before the Frenchman found Teemu Pukki and the striker rounded Martin Dubravka from a wide angle before he set up Dowell, who failed to keep his composure and fired over from close range.
The Magpies nearly opened the scoring just before the half-hour mark as Ex-Canary Murphy swung a left-sided corner in and Dan Burn flicked his near-post header inches wide.
On 35 minutes, Newcastle did find the breakthrough when Murphy played the ball to Allan Saint-Maximin on the edge of the box, who squared it into the path of Sean Longstaff and the midfielder's pass was slightly overhit but Joelinton, who was originally facing his own goal, swivelled and hit a fantastic first-time strike into the top corner.
Before celebrations could begin, the goal had to be checked by VAR for a possible offside on Murphy, but it correctly stood.
And it only took the Brazilian six minutes to double Newcastle's lead as fellow compatriot Guimaraes split open The Canaries backline and as Murphy latched onto it, his shot was superbly blocked, before the ricochet fell nicely for Joelinton and he was left with the simplest of finishes to put his side in complete control.
Howe's side could have been out of sight before the break when Targett's low cross was parried by former Magpie Krul into Longstaff, but he blasted his effort over the bar.
Within just four minutes of the second half, Newcastle ensured they were taking all three points back to the North East, albeit courtesy of a huge error from Krul as the goalkeeper tried to play out from the back, but gifted the ball to Guimaraes and he magnificently dinked it into the net as the boys from Brazil ran the show in Norfolk.
Saint-Maximin almost added a fourth straight from the restart, but his powerful shot from distance was palmed away by Krul.
It took the hosts until the 51st minute before Dubravka was forced into a save as Dowell's 30-yard free-kick was pushed away at full stretch.
Smith opted to make a double substitution in search of a consolation and nearly pulled one back when Jonathan Rowe, who replaced Normann, released Pukki into the box, but the forward sent his effort fractionally wide.
The only slight disappointment to an otherwise perfect afternoon for the travelling Toon Army was when Joelinton was denied his chance of a hat-trick and had to be subbed off after taking a knock to the ankle and made way for Almiron.
And in the closing stages, it could have got even better for The Magpies when Willock saw his deflected effort well saved by Krul and Almiron hit the side-netting.
But following a dominant victory at Carrow Road, the Geordie faithful have now seen their side move into the top-half for the first time this season, which was unthinkable before Christmas and also record 32 Premier League points in 2022 - the second highest in the top-flight behind only title-chasing Liverpool (35).
Norwich destined for the second-tier after suffering a 22nd league defeat of the season
Smith's side offered very little throughout the contest and registered just two shots on target and it's no surprise they have scored the fewest goals in English football this season.
With now just five games left of the season to play, eight points adrift of safety and a -47 goal difference, it's inevitable that Norwich will be relegated.
The Brazilian boys steal the show in the Norfolk sunshine
On a landmark day for Joelinton making his 100th appearance for Newcastle, it also proved to be a memorable one with two goals in the space of six minutes and his display rightfully earned him the man-of-the-match accolade.
His fellow countryman Guimaraes also had a day to remember with a wonderful chipped finish for Newcastle's third and he has now been involved in four goals in his last three appearances for The Magpies as the duo appear to be relishing life under Howe.
Norwich City: Krul; Byram, Hanley, Zimmermann, Giannoulis; Normann, McLean; Dowell, Lees-Melou, Rashica; Pukki. (4-2-3-1)
Newcastle United: Dubravka; Krafth, Lascelles, Burn, Targett; Guimaraes, S. Longstaff, Willock; Murphy, Joelinton, Saint-Maximin. (4-3-3)