Van Persie curled a left-footer into the top corner 20 minutes from time to salvage a 1-1 draw for the Gunners in the Premiership.
Everton had snatched an early lead through Tim Cahill and battled to defend it as if their lives depended on it.
Tim Howard was outstanding in goal for the visitors but will be disappointed not to have kept out Van Persie's equaliser after getting a solid hand to the shot.
• Wenger 'bored' with defensive play
Arsenal have soared up the Premiership table after their faltering start to life in the Emirates Stadium, but their run of five straight Premiership wins ground to a halt as their frustrating inability to convert possession into goals reared its head again.
Arsene Wenger's team may have obliterated Reading last weekend with their fluent football, but they met more rugged and wily opposition in David Moyes' team.
Moyes deployed a five-man midfield in a bid to combat Arsenal's slick passing style and Cahill grabbed the goal to give the Merseysiders hope in the 10th minute.
Everton's free-scoring Aussie arrived to meet an outswinging corner from Mikel Arteta with a trademark leap.
Cahill seemed to loose the flight of the ball in the air and did not connect cleanly with his header.
The ball dropped in front of goal and Cahill reacted brilliantly, firing a shot into the roof of Jens Lehmann's net.
There was some dispute about whether the ball had hit Cahill in the face or on the hand, but the Everton midfielder did not waste any time before smashing the loose ball home.
Everton suddenly had something to defend and they defended gallantly from keeper Howard to Andrew Johnson, who worked tirelessly as the lone striker.
Arsenal felt aggrieved not to get a penalty in the second minute.
Elegant young defender Johan Djourou strode out of defence with the ball and traded passes with Thierry Henry.
Djourou then appeared to be clipped by Phil Neville as he cut through the penalty area, but referee Mike Riley ignored the penalty appeals.
Riley rejected another penalty appeal from the home team in the closing minutes of the first half.
A header from Tomas Rosicky bounced up steeply off the turf and hit Neville near his shoulder.
The first half was interrupted by one flashpoint when Arteta was booked for wrapping a hand around the face of William Gallas and dragging him off the ball.
Gallas reacted angrily, jumping up to confront Arteta and the Arsenal full-back was also booked for dissent.
Everton keeper Howard made a flying save to deny Henry in the 21st minute and produced an even better one to protect the lead in the first minute of the second-half.
Henry thought he had found the equaliser when he climbed high in the goalmouth to head Rosicky's cross down towards the bottom corner.
It looked a certain goal, but Howard sprang to his left and turned the effort around the post with his fingertips.
Moments later and the Everton keeper was beaten by Rosicky's curler from 25 yards, but the effort was inches too high.
Substitute Mathieu Flamini sliced a good chance well wide as Arsenal struggled to convert their pressure into goals.
Flamini was on at right-back for Justin Hoyte, who hobbled off injured just before half-time.
Cahill blocked a ferocious free-kick from Van Persie on the hour and Rosicky rifled the rebound into the side-netting.
Howard saved from Cesc Fabregas and then Henry missed the target with a simple header at the back post.
Wenger sent on Theo Walcott as a raiding right-back in a bid to rescue something. He came on for Djourou and was instantly in the action with a searing burst of pace down the right wing.
He was not directly involved in the equaliser which came just a few minutes later.
Van Persie finally broke Howard's resistance and the Gunners went out for the winner.
Wenger sent on Jeremie Aliadiere and he went close with a header from a corner in the 80th minute.
Everton boss Moyes was sent from the touchline in stoppage time as he argued with referee Riley.
Wenger bored with defensive play
Arsene Wenger claimed he is becoming bored by negative football after watching
Everton hold his Arsenal team at the Emirates Stadium.
Wenger saw the Gunners come from behind to salvage a point thanks to a late
free-kick from Robin van Persie.
Tim Cahill had fired Everton ahead in the 10th minute and the Merseysiders
packed men behind the ball in a bid to protect their lead.
The Arsenal boss was annoyed by David Moyes' tactics and accused the visitors
of time-wasting.
Wenger said: 'Every game we don't win is a massive disappointment.
'It was very frustrating against a team with zero shots on or off target and
when you have 70% of the ball.
'From the first minute to the last the goalkeeper was placing goal-kicks from
the left and then moving them to the right.
'They were standing in front of the ball at free-kicks. It was negative but
that's the game is going at the moment.
'It's like a film you have seen 10 times. It is becoming boring.'
The Arsenal boss said: 'There was no real space and it was not easy up
front.
'We played well but it's not excellent when you cannot score.
'We lacked sharpness and vision in the final third. We didn't attack the ball
well on crosses.'
The Frenchman was also unhappy to see his team leak the early goal from a
corner, handing the initiative to Everton.
Wenger said: 'We mustn't put ourselves in a position where teams are
comforted by that tactic.
'We have to be more rigorous on set-pieces like that and not concede goals.
'But that's easier to correct that if you can't make two passes.
'I have nothing against negative football if it is played with the right
spirit.'
Moyes was sent from the touchline in stoppage time as he gestured to referee
Mike Riley about the amount of time being added.
The Everton boss was waving at Riley and pointing to his watch when the
referee ordered him into the stands.
Moyes said: 'It was added on time. He asked me to leave. I was pointing to my
watch and he took it the wrong way.
'We lost a goal late on against Manchester City and I didn't want it to
happen again.'
Everton did not agree with Riley's decision to award the free-kick, from a
foul by Lee Carsley on Tomas Rosicky, from which Van Persie scored his
70th-minute equaliser.
Moyes said: 'I don't think it was a free-kick.
'It was a gutsy performance, full of character and a different performance to
what we've had this season.
'Our football has been good this season. We've made lots of chances but today
we had to find another way to get a result.
'If we came here to play an open game, they would just cut through us.
'It was tough at times but it was a good draw. Our players deserve a big pat
on the back.
'We were struggling today. We had a bug in the camp. Joleon Lescott was out
of position. We had three strikers on the bench and no midfielders.'
Arsenal 1 - 1 Everton
