The Gunners failed to make the most of their first-half chances and then struggled to create more in the second as visiting goalkeeper Jordan Pickford and his defence held firm.
Arsenal endured a frustrating afternoon in front of goal
Thomas Partey was floored by an awkward tackle from Vitaly Mykolenko
News of Diogo Jota's late equaliser for Liverpool against Fulham was another blow and there were even a few boos at the final whistle.
Arsenal 'keeper David Raya did not have to make a single save.
But at the other end his team lacked penetration and composure.
On a day when the clubs met for the first time since the death of Kevin Campbell, perhaps it was appropriate that it ended honours even.
As early as the sixth minute, Abdoulaye Doucoure had a great chance to put Everton ahead.
But the midfielder took long enough to shoot after being played through by Orel Mangala to allow Gabriel to retreat and make a good block.
From that point on, it was all Arsenal.
Only a combination of poor finishing, Jordan Pickford and great defending, often by James Tarkowski, kept the first half goalless.
Odegaard was guilty of sending one presentable opportunity over the bar and another wide.
Arsenal's corners caused chaos, with Pickford forced into unconvincing punches by deliveries from either side by Declan Rice and Bukayo Saka.
But Pickford produced a fine save in the 29th minute when Saka got the better of Vitalli Mykolenko yet again.
Odegaard eased on to the cutback but when his shot was half blocked and deflected by Tarkowski, the England goalkeeper kept it out with his legs.
Moments later that good work was almost spoiled when Tarkowski's attempted header back to his 'keeper forced Pickford into another panicked punch.
In the dying minutes of the half, an error by Jarrad Branthwaite allowed Arsenal to break but Pickford made a smart stop from Gabriel Martinelli.
The England 'keeper was at it again soon after the break, getting down well to stop a Saka volley after a Tarkowski header fell to the Arsenal man.
Kai Havertz was unable to get a piercing Mikel Merino pass out of his feet and the frustration of the home crowd started to become audible.
Saka continued to have Mykolenko on toast but once more Pickford denied him from a narrow angle after a one-two with Odegaard.
It was no surprise that Arteta made a double change just after the hour as he introduced Jorginho and Ethan Nwaneri. But the identity of the players who went off, Declan Rice and Odegaard, raised some eyebrows.
Everton boss Dyche soon sent on two reinforcements of his own, in Jesper Lindstrom and Armando Broja.
Not much changed. Arteta rolled the dice again with more than 20 minutes to go, sending on Thomas Partey and Gabriel Jesus.
Everton's challenges were starting to get a little more desperate and the first-half siege resumed.
From a free kick, Havertz won a flick on but Gabriel was unable to turn the ball goalwards.
Down the clock ticked. Saka apologised after poorly heading a cross by final substitute Leandro Trossard when Havertz was behind him waiting to connect.
When Partey went down in the box, the home fans screamed for a penalty. But VAR agreed with referee Craig Pawson that Mykolenko's challenge was a fair one.
A minimum of four minutes of stoppage time was indicated.
And Everton saw them out with the minimum of fuss.
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