James Tarkowski netted the only goal of the game to clinch victory for Everton and condemn Arsenal to only their second league defeat of the campaign.
The defender headed home a Dwight McNeil corner just before the hour mark to start the Sean Dyche era with a win, but for Arsenal it was an afternoon of frustration as plenty of good opportunities went begging.
It means that Arsenal miss the chance to extend their lead at the top of the table, while Arsenal winless run at Goodison Park continues.
The opening 45 minutes saw little in the way of goalmouth action for Arsenal, despite dominating the possession stats, with only a weak effort from Thomas Partey giving something for Jordan Pickford to do in the opening exchanges.
Dominic Calvert-Lewin began to find himself at the forefront of their attacks as he headed a corner wide before shooting at Aaron Ramsdale but Arsenal's shot-stopper easily dealt with the danger.
However Arsenal did carve out their best chance of the half on the counter-attack when Buyako Saka slipped in Eddie Nketiah for the first clear sight of goal in the game, but Arsenal in-form striker screwed his effort well wide of the near post.
The big opportunities then all started falling the home side’s way, and after Calvert-Lewin was a stud’s length away from converting Abdoulaye Doucoure’s centre into an open goal, the midfielder looked destined to power home a header a minute later when he was left unmarked in the box but got his attempt all wrong and a great chance went begging.
On a rare foray forward towards the end of the first half, Saka saw a volley beat Jordan Pickford but Conor Coady was in the right place to clear away from his own goalline, but the last word went to the Toffees and Calvert-Lewin was again a whisker away from diverting a cross into the net but the ball dropped wide of the far post.
It was more of the same when the second half kicked off and Ramsdale had to show great reactions to push away a heavily deflected Idrissa Gana Gueye shot from distance within a couple of minutes of the restart, but Arsenal looked dangerous when Arsenal did get forward and after sweeping counter-attack saw great runs by Gabriel Martinelli and Nketiah create a cutback for Martin Odegaard but the skipper uncharacteristically skied his shot.
And that proved crucial as just before the hour mark Dyche’s new team took the lead when McNeil sent a teasing corner towards the far post, and Tarkowski rammed home a header to get Goodison rocking.
With a precious three points now in their grasp, Everton began to sit back and protect them, allowing Arsenal a few more sights of goal. Leandro Trossard saw two efforts from a similar position fail to find their intended target, and Saka was next to waste a good chance when he was found at the back post by Granit Xhaka.
There was plenty more late pressure but Everton were throwing everything at Arsenal, and that meant that a late drive from Oleksandr Zinchenko that whistled over the bar was Arsenal's best late chance of note as Arsenal failed to score in a Premier League game for just the second time this campaign.