The north London side headed into the game looking for a win to move up into third place in the Premier League table as the race for Champions League qualification hots up.
However, the Gunners failed to click into gear and Everton took the lead in the 10th minute when Phil Jagielka fired home after poor defending from the Gunners.
Arsenal were not able to conjure many chances for an equaliser and ended up slumping to their seventh defeat in the Premier League this season.
The result left Arsenal in fourth place in the table, but Chelsea FC can push them out of the top four if they beat West Ham United on Monday night.
And former England star Lineker was not particularly impressed by what he saw from the Gunners at Goodison Park on Sunday afternoon.
Posting on Twitter after the game, Lineker said: “Good performance from @Everton who should have won comfortably. @Arsenal really poor on the road again.”
Arsenal will return to Europa League action on Thursday night when they take on Napoli in the first leg of their quarter-final clash at The Emirates.
The Gunners travel to Watford in their next Premier League game on Monday 15 April.
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