The Gunners could not have wished for a better start to the game, taking the lead inside 60 seconds when Theo Walcott fired home from a tight angle after exchanging passes with Aaron Ramsey.
However, almost immediately, the visitors suffered a blow when Laurent Koscielny appeared to pull a muscle and had to be replaced by Kieran Gibbs.
The early change seemed to knock Arsenal out of their stride and Everton almost grabbed an equaliser when Steven Pienaar had a shot blocked after being set up by Marouane Fellaini before the Belgian forced Wojciech Szczesny to make a smart save.
Szczesny then had to keep Fellaini's downward header out but the Everton midfielder drilled home a 28th-minute equaliser after a series of sloppy errors by Arsenal.
David Moyes' side almost took the lead five minutes before the interval but Nikica Jelavic smashed his shot over the bar after the lively Fellaini had flicked the ball on.
The second half was disappointing as two evenly-matched teams cancelled each other out for long periods. However, Gibbs did well to stop Steven Naismith getting on the end of Jelavic's cross and Mikel Arteta appeared to be fortunate not to concede a penalty when the Spaniard pulled Pienaar back inside the area.
For Arsenal, Olivier Giroud flashed a header from Walcott's cross narrowly wide but neither side could snatch a late winner as two of the Premier League's draw specialists had to settle for a point once again.