The Gunners wasted several clear-cut openings, most notably through Olivier Giroud, in a pulsating encounter that did neither side's Champions League prospects much good
The Gunners remain in third place, two points ahead of Chelsea albeit having played two games more, while David Moyes's side lie a further two points adrift.
The visitors, who impressed hugely on their last visit to north London in an ertaining 2-2 draw with Tottenham earlier in April, were dominant in the early stages, pressing high up the pitch and denying Arsenal's artisans time and space to create.
In possession, meanwhile, Everton carried a significant threat and should have taken a sixth minute lead when Seamus Coleman's pass dissected the home defence, only for Steven Pienaar to display an uncharacteristically rash touch in front of goal, firing high over Wojciech Szczesny's crossbar.
What the game lacked in aesthetics, it more than made up for in brute force as Everton set about their business in typically bruising fashion. Marouane Fellaini was guilty of several early transgressions that might otherwise have resulted in cautions, while Darron Gibson was fortunate to remain on the field following two bone-shuddering challenges on Theo Walcott in quick succession.
Arsenal's threat had come down their right-hand side and it was from one enterprising foray that they came closest to breaking the deadlock in the opening half.
Aaron Ramsey escaped the attentions of Leighton Baines and whipped an inviting low ball across the six-yard box which was met by Olivier Giroud on the slide – the French striker was, however, unable to provide the finish that would have given his side an undeserved lead three minutes before the interval.
The feisty nature of the contest continued after the half-time whistle with Kevin Mirallas and Jack Wilshere involved in a tunnel altercation that saw the Belgian squirt the contents of his water bottle over the England midfielder.
The 21-year-old, starting his second game in three days following a five-and-a-half week absence, was struggling to influence the game but his midfield partners, Santi Cazorla and Ramsey, quickly warmed to the challenge in the second half, with the latter testing Tim Howard with a fizzing drive from 25 yards.
At the other end Baines, unusually for him, was wasteful with a free kick that appeared tailor-made for him, while the increasingly influential Ross Barkley was inches wide with a rasping drive from 20 yards that was shaved Szczesny's left-hand post.
With 15 minutes left of a potentially season-defining game for both sides, both managers remained commendably committed to attack and it would have paid dividends for Arsenal had Giroud not been prevented by an astonishing goal saving clearance from Sylvain Distin, before he blazed over soon after when a more controlled finish would surely have broken the deadlock.
In the end, a point represented a fair outcome, one that will be celebrated by both Chelsea and Spurs in a race for a top finish that assumes a different and more fascinating complexion by the game.