The Arsenal captain had cancelled out Frank Lampard’s opener in the first half before John Terry put the home side ahead on the stroke of half-time.
But the Gunners responded in emphatic fashion immediately after the break as Andre Santos equalised and Theo Walcott fired ArsÃ¨ne Wenger’s men into the lead.
Chelsea drew level with 10 minutes to go through Juan Mata – but Van Persie’s late double sealed a famous win for the Gunners, who move up to sixth in the Premier League table.
Arsenal started the brighter of the two sides and could have twice taken the lead, but Gervinho scuffed his shot wide after being found by Walcott’s low cross. And moments later Van Persie fired over the bar from close range.
The Gunners were soon made to rue the missed opportunities when Lampard nodded Chelsea ahead in the 13th minute, heading Mata’s pinpoint cross past Wojciech Szczesny and into the bottom corner.
Despite falling behind, the visitors ramped up the pressure and they carved out an exquisite equaliser in the 35th minute. Aaron Ramsey slid in Gervinho, who raced through on goal before squaring the ball for Van Persie to tap into an empty net.
In a game full of twists and turns, the home side then took the lead again just before the break. Terry got ahead of Per Mertersacker and bundled home Lampard’s corner on the stroke of half-time.
But amazingly, the Gunners were level again just four minutes after the break. Alex Song picked out Santos with a searching low pass and the Brazilian slotted his shot under Petr Cech in the 49th minute.
The game was now incredibly open – and just moments after Arsenal equalised referee Andre Marriner had to make a potentially game-changing decision when Szczesny raced out and brought down Ashley Cole on the edge of the box.
However, the referee decided there were enough red shirts back to ensure it was not a clear-cut goalscoring opportunity and booked the Arsenal shot-stopper, who pushed the resulting free-kick round the post.
Walcott then put the Gunners in front for the first time in the afternoon. The England winger slipped past Cole and coolly fired the visitors into the lead after 55 minutes.
But with 10 minutes left, Mata struck to pull Chelsea back level when he sent a fantastic effort sailing past Szczesny and into the top-left corner after capitalising on a sloppy pass from Santos.
However, it was the visitors who had the final say five minutes later. A risky pass back from Florent Malouda to Terry invited pressure from Van Persie, and as the England captain slipped, the Arsenal striker burst through unchallenged, rounded Cech and slotted home to put Arsenal 4-3 up and spark wild celebrations.
Incredibly, there was still time for more. The Gunners caught Chelsea on the break and Mikel Arteta slid in Van Persie, who sent a thumping strike past Cech to complete his hat-trick and decide a wonderfully entertaining game in west London.
“It was fantastic to watch, both teams went for it. In the second half I felt we were completely dominating the game and created chance after chance.
“Overall we have shown quality and mental strength as well. To go 2-1 down and come out the way we did shows great resolve in the team.
“I thought we looked fragile in defence in the first half but we looked dangerous going forward.
“We achieved the target of winning a big game away from home and did it with style, but we have to keep our feet on the ground and focus on another big game on Tuesday.
“That’s the football we want to play, you always think you can get rid of the defensive mistakes we had in the first half.”
“The decisive moment might have been when Mata was free on goal and misses the chance to go 4-3 ahead.
“We’ve seen some great open games recently between the top teams and this is one more. We will have to reflect on the mistakes we made and there are a couple of things we can improve, but the commitment and desire is there.
“We must evolve, it’s a loss against a big team picking up form, it’s two defeats in a row and we have to get things right.”
Chelsea: Petr Cech, Ashley Cole, Branislav Ivanovic, John Terry, JosÃ© Bosingwa, Frank Lampard, Ramires (Romelu Lukaku, 72), John Obi Mikel (Raul Meireles, 76), Fernando Torres, Daniel Sturridge (Florent Malouda, 62), Juan Mata.
Arsenal: Wojciech Szczesny, Johan Djourou (Carl Jenkinson, 76), AndrÃ© Santos, Per Mertesacker, Laurent Koscielny, Theo Walcott (Tomas Rosicky, 79), Aaron Ramsey, Mikel Arteta, Alexandre Song, Gervinho (Thomas Vermaelen, 88), Robin van Persie.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news