Bendtner got his ninth goal of the season for Arsenal
Arsenal striker Nicklas Bendtner scored the winner to leave Everton still waiting to seal a European place.
Bendtner headed home from an Armand Traore cross late in the game to secure a third straight league win.
Arsenal were not at their best but Emmanuel Adebayor and Gilberto had good chances before Bendtner's winner.
Andy Johnson and Joleon Lescott both went close for Everton, who require a point to secure fifth position and a place in the Uefa Cup.
The defeat means Everton will face an anxious last day of the season, when they host Newcastle needing a draw to be sure of another European campaign.
Wenger praised his 'outstanding' players
Arsenal saw any lingering hopes of the title end with Manchester United's win on Saturday, but they can still pass Chelsea for second place.
The two sides combined to produce the sort of game that will have been more to the liking of Everton manager David Moyes than opposite number Arsene Wenger - even if the result did not go their way.
Arsenal were allowed plenty of possession but only the briefest of sights at goal.
Theo Walcott curled a shot wide and Gilberto headed off target after an opening 20 minutes that had end-of-season affair written all over it.
Johnson gave Arsenal a rare scare after racing on to a Manuel Fernandes through-ball, but his effort was blocked by Lukasz Fabianski's legs and Denilson cleared Lescott's header off the line from the resulting corner.
Emmanuel Eboue saw an effort deflected wide, while Bendtner, Gilberto and Adebayor all had headers that failed to test Tim Howard in the Everton goal.
With Cesc Fabregas watching from the stands on his 21st birthday, Arsenal were lacking their usual drive from midfield and too often opted to try and pick out an isolated Adebayor, who was well marshalled by Phil Jagielka and Joseph Yobo.
Interview: Everton boss David Moyes
Everton threatened to snatch a first win at Arsenal since 1996 when Lescott found room to shoot in the box but flashed his left-foot effort wide.
Jens Lehmann was given a run out and received a rousing reception from the home fans.
The keeper is one of a number of players who may not be around next season.
But Bendtner boosted his hopes of being a firm part of Wenger's plans for next season by scoring the winner with 13 minutes remaining.
He drove down the middle and slipped a pass in for Traore, whose left-foot cross was perfect for Bendtner to leap and thump past Howard.
Arsenal manager Arsene Wenger
"Overall we had a big disappointment because we didn't win the league but we were so close.
"Like every winner we will aim to come back next season and do it.
"It's very encouraging with so many players out we were still strong at the back."
Everton manager David Moyes
"There is a lot of disappointment not to get the point but the performance was good, the players stuck at it. It is always difficult here because Arsenal have so much possession.
"I thought things had started to turn in our favour but they are a very good counter-attacking team.
"We have been fourth or fifth all year and we want to hang on to it."
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