Chelsea had taken a first-half lead through an own goal by Johan Djourou but the west Londoners, who had not conceded a second-half league goal all season, succumbed to the Dutchman's brilliant finishing.
The Blues remain at the top of the league table but Arsenal are now closing in and are now serious title challengers.
Chelsea supplied the early pressure with a problematic offside and a possible penalty both going against the home side.
Arsenal countered with a period of excellent possession forcing Chelsea to concede a couple of corners.
One attack saw Petr Cech spill a 12-yard shot by Cesc Fabregas on 13 minutes The Spanish midfielder followed this up a minute later with a vicious right-footer - but this time Cech was equal to the effort.
Denilson was next to pepper the Chelsea goal, on 15 minutes, but this time his shot was high and wide of Cech's goal.
Frank Lampard beat the Arsenal defence on 23 minutes, passing the ball to Jose Bosingwa on the right, before continuing his run.
The Portugal right-back then sent in a curling ball which the onrushing Lampard met with his head. It was powerful enough to force Manuel Almunia to make a smart save.
The home side made the breakthrough on 30 minutes when a quick throw-out by Almunia resulted in a loss of possession by Arsenal.
The ball was fed to Bosingwa on the right. His inch-perfect cross into the Arsenal area led to Djourou poking his right foot out and redirecting the ball into his own net from six yards out.
Chelsea continued to dominate possession early in the second half and should have doubled their lead in the 51st minute. A quick break saw the ball fall to Lampard just outside the Arsenal area, but the midfielder scuffed a left-footed effort wide of the goal.
But Arsenal, with their first attack of the half, levelled on 59 minutes. From a clear offside position van Persie collected a square pass from Denilson just outside the Chelsea area.
The Dutchman took just one step forward before blasting a right-footer past a helpless Cech.
Two minutes later and Arsenal were ahead. A long ball by Gael Clichy found Emmanuel Adebayor who headed down to van Persie. He swivelled and unleashed a low left footer across Cech's body and into the left-hand corner of the net.
Chelsea boss Luiz Felipe Scolari brought on winger Florent Malouda for Mikel on 69 minutes.
The Frenchman's first taste of the action was to send a free-kick high into the seating behind Almunia's goal on 77 minutes.
Scolari introduced debutant Miroslav Stoch on 81 minutes for Deco. However, he didn't touch the ball and Chelsea have now dropped 12 league points at home.