The Gunners had won four league games in succession to move above both Tottenham and Chelsea in the race for Champions League qualification.
However, despite plenty of endeavour, Arsene Wenger's men were unable to breakdown a determined Everton side, who utilised all their physical qualities if not collecting the win they probably needed to harbour any real hopes of getting into the top four.
Arsenal's best chances fell to Olivier Giroud, but the France striker failed to test Tim Howard and the hosts had to be content with a point which sees them move two clear of their London rivals below them, but having played more matches.
Gunners midfielder Jack Wilshere started his second match in four days since returning from an ankle injury, while England forward Theo Walcott came into the side, but captain Thomas Vermaelen dropped to the bench with Per Mertesacker returning from a one-match suspension.
Both teams sported black armbands in remembrance of the Hillsborough disaster and also Monday's tragedy in Boston, with a period of silence impeccably observed before kick-off.
It was a lively start by Everton, who had won four of their last five matches.
After six minutes, Steven Pienaar was played into the right side of the Arsenal penalty area by Seamus Coleman, but lifted his shot over both goalkeeper Wojciech Szczesny, back in the side because of injury to Lukasz Fabianski, and also the crossbar.
A few hefty challenges then went in from both sides as the midfield battle heated up.
Wilshere was left in a heap by a strong sliding tackle from Ross Barkley, the 19-year-old drafted into the Everton starting line-up on Tuesday night by David Moyes.
The England midfielder, who had been out for six weeks with an ankle problem, was, though, soon back up and running.
Arsenal pressed again on 25 minutes, the ball worked out to full-back Kieran Gibbs on the overlap down the left, but his angled drive flew into the Clock End.
Referee Neil Swarbrick finally reached for his pocket to show Darron Gibson a yellow card after the Everton midfielder clattered into Walcott.
Gibson was fortunate to stay on the pitch when he bodychecked the England winger a few minutes later, the referee instead deciding just to dish out a stern talking too - much to the disbelief of Gunners boss Wenger.
The notebook was, though, out again when Pienaar blocked Walcott's run towards the right side of the Everton penalty area.
Chances were few and far between, as Victor Anichebe squirmed a shot away in the Arsenal six-yard box, which hit a defender and then Szczesny's legs before being gathered.
On 41 minutes, Giroud should have put Arsenal ahead when Aaron Ramsey got away down the right and swept the ball back through the penalty area, where the French striker arrived at pace ahead of goalkeeper Howard - but stabbed the ball wide.
Santi Cazorla's angled shot was blocked as the Gunners finished the half strongly.
The Spaniard tested Howard again following the restart with a rasping half-volley as Arsenal looked to break down the stubborn visitors.
Everton, though, refused to lie down and were awarded a free-kick just outside the penalty arc in a central position when former Toffees midfielder Mikel Arteta tripped Pienaar - but Leighton Baines fired it into the wall.
From the break, Walcott sprinted away down the right and eventually crossed for Giroud, but with his back to goal, the striker could not turn and the chance was gone.
Barkley was inches away from slotting Everton ahead with a curling effort after sidestepping a couple of challenges on the edge of the Arsenal penalty area.
With 20 minutes left, Wenger made a double change as Wilshere was replaced by Alex Oxlade-Chamberlain and Lukas Podolski came on for Walcott.
Needing victory to close back on the top four, Everton sent on Croatian Nikica Jelavic for Barkley.
Chamberlain darted away down the right on the overlap, but his cross into the area was booted clear by Coleman with Giroud ready to tap in - although the France forward looked to have strayed a yard offside.
Giroud then showed good footwork to fashion a shooting chance from a tight angle, but blasted over.
Arsenal pressed during the closing stages, but unlike here against Norwich on Saturday, there was to be no late surge and the points were shared.