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.
Both sides have hopes of playing in the Champions League next season, but Arsenal will be left to rue the missed chances after carving out opportunities in both halves for French international striker Olivier Giroud.
''We should have won the game, but it keeps us in a strong position for the top four,'' Arsenal manager Arsene Wenger said.
Everton midfielder Ross Barkley blazed a shot narrowly wide in the second half of a closely-contested game at Emirates Stadium.
''I would take a Europa League place now, but I'm going for the Champions League,'' Everton manager David Moyes said. ''We more than thoroughly deserved a point but we did not do enough to win it.''
Both teams wore black armbands to honor the victims of the Boston Marathon bombings and commemorate the 24-year anniversary of the Hillsborough disaster. A moment's silence was also held before the game.
In a physical and feisty first half, Steven Pienaar had the first good opportunity after just five minutes when he found the ball at his inside the six-yard box, only to see his effort sail harmlessly over the bar.
Arsenal gradually took charge of the game but missed two good chances on the stroke of halftime. Aaron Ramsey charged down the right flank and sent a cross in toward Giroud, who reached the ball ahead of the onrushing Tim Howard but sent the ball wide. Santi Cazorla then charged into the box moments later only to his effort acrobatically diverted for a corner from the sliding Phil Jagielka.
Arsenal started brightly after the second half, with Kieran Gibbs looking increasingly dangerous down the left hand side.
He first set up Cazorla for a volley from the edge of the area that forcing Howard to push the ball away for a corner. Gibbs then stormed into the edge of the area minutes later but his cross could not find anyone in a red shirt.
The impressive Barkley almost opened the scoring in the 64th, his curling effort just evading Szczesny's right-hand post. But that was the visitors' only chance in a second half dominated by the home side.
With 10 minutes left, Arsenal then produced the best move of the game as Gibbs was expertly fed the ball by the industrious Cazorla and sent a low cross in toward the stretching Giroud, but Seamus Coleman managed to slide in to clear.
Giroud had another great chance three minutes later as he received the ball in the area and went past Phil Jagielka only to blast his effort over the bar from a tight angle.
''Maybe we used too much power instead of technique in our finishing,'' Wenger said. ''But you have to respect Everton for the effort they put in.''