Tim Cahill had earlier scored his 100th career goal which looked to have secured maximum points for the home side.
The makeshift striker had been having a quiet game but, on the hour mark, he latched on to Leighton Baines' pinpoint cross to power a header past the hapless Manuel Almunia and send Goodison Park into raptures.
But they were silenced with only a minute left when van Persie rifled the ball past Tim Howard and into the far corner of the net.
The home side thought that victory was assured and the result will be a big dent to their hopes of claiming a UEFA Cup spot.
The away side dictated play from the kick-off and it took Everton five minutes to get the ball in the opposition's half.
Arsene Wenger's men could have taken the lead as early as the second minute when Steven Pienaar fouled Bacary Sagna on the edge of the Everton area, but van Persie's free-kick was deflected away for a corner which came to nothing.
Everton soon started to find their rhythm though and would have scored from their first meaningful attack of the game after 13 minutes had the ball not been taken away from the feet of Marouane Fellaini by William Gallas with Almunia stranded.
Instead the first half was full of endeavour but lacked quality, especially in front of goal.
Both sides continued to press for the opener but it was Arsenal who had the better chances with van Persie and Emmanuel Adebayor going close, before Howard watched Denilson's 20-yard effort whistle wide of his near post.
The home fans were eager for a touch of deja-vu three minutes before the break, but Mikel Arteta's drilled free-kick was comfortably cleared by the Gunners defence - a feat Liverpool had failed to do two weeks ago.
The second half started better but Denilson really should have put Arsenal 1-0 up on 52 minutes.
The menacing Adebayor let the ball run through to the unmarked midfielder only for the Brazilian to blaze over the bar and leave manager Wenger fuming on the sidelines.
They were made to pay when Cahill headed past Almunia after Baines' brilliant run and cross from the left wing.
Buoyed by their goal, Everton took the game to Arsenal and Cahill could have got his second of the night had he not hit his shot straight at the Gunners keeper.
But with only seconds remaining van Persie was left unmarked at the back post and he struck a thunderous effort past Howard to earn Wenger's men a point.