The astonishing victory was Arsene Wenger's 500th since he became Arsenal manager in 1996, Arsenal's first away win in the league this season and Chelsea's first home defeat as the English top-flight continues to dish out goals galore.
In the other early game, Manchester United bounced back from their harrowing 6-1 defeat by Manchester City last weekend when they beat Everton 1-0 away with a first-half goal from Javier Hernandez.
The win moved second-placed United to within two points of leaders City, at least until their neighbours play Wolverhampton Wanderers in a 1400 GMT kickoff.
While United were involved in the most dramatic game of the day last weekend, they had to concede top billing to Chelsea and Arsenal on Saturday in another hugely entertaining Premier League clash.
Chelsea led through Frank Lampard after 14 minutes before Van Persie scored his first in the 36th minute and Chelsea skipper John Terry took advantage of some woeful Arsenal defending to put his side 2-1 up in the last minute of the first half.
Andre Santos struck Arsenal's second equaliser early in the second half before Theo Walcott scored a brilliant solo goal, picking himself up and continuing his run after slipping over, to give Arsenal a 3-2 lead after 55 minutes.
Juan Mata then equalised for Chelsea with a brilliant swerving shot, which Arsenal goalkeeper Wojciech Szczesny had no chance of saving, to make it 3-3 after 80 minutes.
Chelsea then committed suicide five minutes later with some more poor defending with Florent Malouda hitting a stray pass back to Terry, who stumbled to leave Van Persie to run in on Chelsea keeper Petr Cech and blast the ball into the gaping net.
The Dutchman then made it 28 goals from his last 27 Premier League games in this calendar year to make it 5-3 in the second minute of stoppage time with a swerving shot of his own to match Mata's.
Arsenal, who made a poor start to the season, climbed up to sixth place until the later matches at least and celebrated as if they had won a cup final.
Van Persie told Sky Sports: "You could see how happy we were at the end. We fought so hard for it, every single one of us -- it's a big, big win for us."
Wenger, whose team have been dismissed as title challengers following their poor start, said: "We needed to win with style and class and we did that against a big team and it was a big win for us.
"In the first half we looked fragile in defence, but less so in the second and kept going forwards and that made all the difference."
The result consigned Chelsea to a second successive defeat in a London derby after they lost at Queens Park Rangers last weekend and was the first time they had conceded five at home in a league match since Liverpool beat them 5-2 in the old first division in December 1989.
It was also the first time Arsenal had scored five against Chelsea since a 5-2 win over them in 1979 and Chelsea's stunned manager Andre Villas-Boas said: "It was so open it could have gone either way. We will have to reflect on the mistakes we made -- there are some things we must improve."
At Goodison Park, Hernandez scored what proved to be the winner and gave United their first victory at Everton since September 2007 when he capitalised on good approach work from Tom Cleverley and Danny Welbeck before Patrice Evra crossed from the left for the unmarked Mexican to turn in on 19 minutes.
United manager Alex Ferguson told the BBC: "It's an important result to come away from home and win.
"Everton is never an easy game and they put everything they could into the game. First half we played some fantastic football and scored a good goal. Hernandez is fantastic in the box. It's a good result and we worked hard for it."