Grant Holt and Simeon Jackson scored as Norwich beat Aston Villa at Carrow Road to end their fine Barclays Premier League campaign on a high.
The Toffees opened the scoring in the 16th minute through Steven Pienaar's deflected effort and went 2-0 up in the 27th when Nikica Jelavic finished smartly at the second attempt after goalkeeper Tim Krul had blocked his initial shot.
Johnny Heitinga then netted a 65th-minute header and although Tony Hibbert scored an own-goal eight minutes later, it proved little more than a consolation for Newcastle, who finish outside the Champions League spots in fifth.
Everton - who had substitute Tim Cahill sent off for his part in a mass skirmish after the final whistle - have matched their effort from last season in claiming seventh place, and their supporters will enjoy the additional bonus that it is higher than their Merseyside rivals Liverpool.
Newcastle fans, meanwhile, will be disappointed their team will not be playing Champions League football next term, but with it being something which depended on other results today and potentially the outcome of the final of this year's competition as well, they will not be too down about the team's achievements.
Finishing as high as they have only two years after being promoted back to the top flight and with the Europa League to look forward to, the Magpies and their boss Alan Pardew - who was named Barclays Premier League manager of the season earlier this week - can reflect with great satisfaction on a campaign few could have anticipated before it got under way.
With Joey Barton, Kevin Nolan and Jose Enrique leaving the club and Andy Carroll having already departed, expectations among Newcastle supporters were less than sky-high last summer, but some of the players Pardew has brought in have proved masterstrokes, the likes of Papiss Cisse, Demba Ba and Yohan Cabaye rapidly developing into Toon icons over the course of the campaign.
In today's final-day clash, though, they were ultimately outshone by their opponents.
Newcastle almost went in front in the opening minute when a free-kick ricocheted around the Everton box, but Tim Howard dived on the ball to prevent it bouncing across the line and after Ba had fired a shot well over, the hosts began to get into their stride.
Marouane Fellaini sent a header off-target and five minutes later, the Toffees had the lead when the Belgian midfielder teed up Pienaar, whose strike from outside the box took a deflection off Magpies defender Mike Williamson and looped in.
Pienaar almost had another in the 21st minute but rifled narrowly wide after the ball had come across goal to him from Leon Osman.
Everton kept pushing forward and just before the half-hour they did double their advantage through their in-form marksman Jelavic.
Having brought the ball down - possibly with his arm - and unleashed one shot which Krul saved, the Croat was alert to tuck away the rebound for his 11th goal since joining the Merseyside club in January.
The hosts were mounting attack after attack, with Krul just beating Fellaini to a cross and Darron Gibson and Osman shooting wide.
Gibson then sent another effort down Krul's throat, with Newcastle's only reply in among the bombardment being a Jonas Gutierrez attempt which whistled the wrong side of the post.
The visitors managed to apply some pressure right at the end of the first half, but Cisse shot wide and Hatem Ben Arfa missed the target with a free-kick.
Cisse dragged a shot wide moments after the restart, but Everton were swiftly back on the offensive, with Jelavic turning and firing off-target.
Gibson brought a save out of Krul with a free-kick and after Howard had got in the way of a Ba strike, the Toffees added a third goal.
Heitinga was the man to head it home, connecting with another Gibson free-kick.
The job looked done for the hosts, but then Hibbert got in a mix-up and headed past his own goalkeeper.
It threw Newcastle a lifeline, and Howard then did well to claw away a header from Cisse.
The visitors could offer no more, though, and Everton might have had a fourth by the end, with Cahill being denied by Krul, Gibson shooting wide and Osman hitting the post.
There were disappointing scenes after the final whistle as players from both sides got involved in a skirmish, with Cahill being shown a red card.