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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