The Londoners, who had won four in a row to move up to third and are chasing qualification for the continent's elite competition for a 16th straight season, had the better opening with French striker Olivier Giroud guilty of two bad misses.
Giroud prodded wide at full stretch with the goal at his mercy late in the first half and blazed over from a good position near the end to cap a frustrating night for the hosts who were put off their stride by a well-organised and combative Everton side.
"The game was basically down to physical intensity, it was unbelievable from first to last minute," said Arsenal manager Arsene Wenger.
"The referee let a lot go unpunished and in the end we put a lot of effort in. Maybe we used too much power instead of placing the ball.
"The finishing was missing. I think (Everton midfielder) Darron Gibson could have been sent off. He got away with a few tonight. We are in a strong position but we have to win our next game. If we keep this spirit I'm sure we will get there."
Arsenal have 60 points from 33 games followed by Chelsea on 58 from 31, Tottenham Hotspur on 58 from 32 and Everton with 56 from 33.
Leaders Manchester United, with a 15-point advantage over Manchester City, can move closer to regaining the title when they play at West Ham United on Wednesday when City host fellow FA Cup finalists Wigan Athletic while Chelsea visit Fulham.
Wenger cut an annoyed figure throughout the first half on Tuesday, spending much of the time complaining to the fourth official at what he deemed to be some over-zealous challenges.
Gibson was booked and then escaped with a lecture, to the howls of derision from Arsenal fans baying for red, after appearing to cynically bodycheck Theo Walcott.
Everton, who have not won at Arsenal since 1996, had clearly set out to stop the England frontman and Steven Pienaar was shown yellow for a clear shove to stop Walcott's progress towards the penalty area.
Pienaar had wasted a good early opening when he fired over from six metres from Seamus Coleman's low cross behind Arsenal's back four.
Arsenal, unsettled by the visitors, countered with Kieran Gibbs's skied effort and finished the half on top.
Giroud was left to wonder how he missed when he stretched to meet Aaron's Ramsey's cross but could only divert wide with the goal gaping and a well-timed Phil Jagielka block denied Santi Cazorla.
Arsenal pressed strongly after the break with Tim Howard beating away a fierce Cazorla strike.
Everton were always a threat on their rare forays forward and impressive young midfielder Ross Barkley curled a 20-metre effort inches wide.
Manager David Moyes said his side faced a tough task to seal a top four place.
"The games are running out now so it's harder and harder for us to get into the top four," he said. Probably we will get talked about at the end of conversations about the Champions League as we are rank outsiders. But there's no shame in getting a draw here."