Alan Pardew's side needed to collect three points while hoping results involving Arsenal and Tottenham elsewhere fell in their favour, but were undone by their own sloppy performance, especially in the first period when they seemed to take leave of their senses.
Newcastle had kept five clean sheets in eight games heading into this match, but their rearguard was far from composed as the home side struck twice in the opening 27 minutes.
Steven Pienaar drove a shot beyond visiting goalkeeper Tim Krul via a wicked deflection off Michael Williamson on 16 minutes before Jelavic gave Fabricio Coloccini the slip to add the second 11 minutes later.
Pienaar drove an effort off a post on 21 minutes after chaos in the visiting area, but David Moyes's team were always looking the likelier lot as they nailed a second goal.
Krul produced a magnificent save from former Rangers man Jelavic who had appeared to use his arm to control the ball, but the Croatian striker kept his composure magnificently to rattle the rebound into the net.
The unmarked Johnny Heitinga headed the third goal from Darron Gibson's free-kick on 65 minutes before Tony Hibbert's own goal on 73 minutes completed the scoring giving the visiting side some consolation from a pretty forgettable day.
Newcastle altered matters at half-time as James Perch and Davide Santon departed for Sylvain Marveaux and Ryan Taylor. They at least looked a bit more sprightly in the second half.
Everton goalkeeper Tim Howard - who was making his 183rd straight performance for the home side - appeared to pick up a leg injury in the second half yet still made decent saves from Papiss Cisse and Demba Ba while Gibson was unfortunate not to net having thrashed a couple of fine efforts at the visiting goal.
The game ended amid some poor scenes with Everton substitute Tim Cahill grabbing the throat of Yohan Cabaye - who was forced to apologise to a ball boy after pushing him to retrieve the ball in the first period - in injury time. He was sent off in disgrace, but Everton had long since pocketed the three points.
Newcastle were hoping to become the first team from outside Manchester, Liverpool or London to finish the season in the top four in 11 years, since Newcastle themselves finished third in 2002-03.
In the final analysis, Newcastle finish in a respectable fifth place with Tottenham completing a 2-0 win over Fulham to snatch fourth spot and third-placed Arsenal winning 3-2 at West Brom.
They will compete in the Europa League next term while Everton finish seventh in the table after a fourth straight win. They finish four points ahead of Merseyside rivals Liverpool, who ended their season with a 1-0 loss at Swansea.