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.
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.
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.
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."
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."
