The Gunners remain third in the table, two points ahead of Chelsea and Tottenham, but having played two games more than the Blues and one more than Spurs, with Everton two adrift of the White Hart Lane club in sixth.
Everton had a good chance to open the scoring with only seven minutes on the clock when Steven Pienaar burst into the box following Phil Jagielka's incisive pass, only to lift his shot beyond Wojciech Szczesny but also over the crossbar.
It took 24 minutes for Arsenal to muster their first shot, but even then Kieran Gibbs' effort from outside the box after being set up by Santi Cazorla following Marouane Fellaini's poor clearance failed to trouble Tim Howard.
There was plenty of bite in the tackles from both teams with Darron Gibson going into the book for a challenge on Theo Walcott in the 28th minute and very fortunate not to see red just six minutes later for a blatant body check on the same player.
Referee Neil Swarbrick's decision-making then came under the spotlight even more when he cautioned Pienaar for an almost identical offence on the England international, with his inconsistency continuing as Olivier Giroud escaped any retribution for going through the back of Seamus Coleman as he shepherded the ball out.
After a largely uninspiring opening 40 minutes, there was a flurry of opportunities as the interval approached, with Ross Barkley playing in Victor Anichebe, who saw his initial shot blocked with Szczesny grateful to grab the ricochet as it rolled behind him.
Giroud really should have done better when presented with a glorious chance from Aaron Ramsey's teasing cross from the right but could only fire wide, while Cazorla was denied by a superb sliding challenge from Jagielka after nut-megging Gibson in the area.
There was an unseemly spat between Kevin Mirallas and Jack Wilshere as the teams headed for the tunnel at the interval, with the Gunners finally getting a shot on target after the break as Gibbs' knock-down was fired powerfully goalwards by Cazorla but Howard was equal to the challenge and fisted the ball away at his near post.
Anichebe could have broken the deadlock in the 54th minute but he failed to get a clean contact on Leighton Baines' curling free-kick from the left, while the England left-back fired another opportunity into the wall, with Arsenal immediately breaking only for Giroud to waste Walcott's good work.
Barkley was inches away from a spectacular opener as he curled a shot towards Szczesny's top corner in the 65th minute only to see it fly agonisingly wide with the goalkeeper beaten, with Arsenal then seeing two half-chances in quick succession for Ramsey come to nothing.
Arsenal looked as though they had opened Everton up on the counter in the 78th minute as substitutes Lukas Podolski and Alex Oxlade-Chamberlain combined, but the latter's pass in towards Giroud was superbly kept out by a combination of Coleman and Sylvain Distin, with Giroud blazing over when well-placed moments later.