The second of van Persie's goals was a wonder strike as he acrobatically volleyed Emmanuel Eboue's cross from the right into the back of the net giving Charlton keeper Scott Carson no chance.
Gunners manager Arsene Wenger, who this week celebrated 10 years in charge of the North London club, hailed the goal as one of the best he had ever seen.
"Nobody expected it," Wenger said.
"When the ball came over I thought we didn't have anyone in the box and then suddenly van Persie came in at full pace and power.
"I thought it was going to go over the bar but the technique was perfect. It was the goal of a lifetime and definitely one of the best I have seen since I took charge at Arsenal.
"I don't know whether it is the best - I have seen some good ones since I came but it is certainly in the top five. It was unbelievable."
Charlton took the lead against the run of play in the 21st minute when striker Darren Bent tapped in for the home side from Jimmy-Floyd Hasselbaink's cross after the latter had been close to offside in the lead up to the goal.
However, the visitors hit back when van Persie scored his first, latching on to a pass from midfielder Alexander Hleb and rifling home from inside the box.
The loss kept Charlton at the foot of the Premier League table after seven matches, with new manager Iain Dowie under pressure after his side's disappointing start to the season.
"We have now lost six out of seven and I am responsible for that," Dowie said.
"We had opportunities and the effort and the commitment were excellent. We have to make sure we bounce back."
Chelsea held at home
Meanwhile across London at Stamford Bridge, Chelsea were held to a 1-1 draw by the undefeated Aston Villa as Didier Drogba's early goal for the Blues was cancelled out by Villa striker Gabriel Agbonlahor's header late in the first half.
Visiting manager Martin O'Neill spoke highly of his 19 year-old goal scorer who holds the Aston Villa youth team record for most goals in a season by netting 40 times in 2003/04.
"I thought Gabriel Agbonlahor was fantastic. I knew very little about him when I joined the club but he has been a breath of fresh air," O'Neill said.
Villa's Gabriel Agbonlahor (R) rose
to the occasion against Chelsea
"He is exciting and wants to take players on and get at the opposition."
Chelsea manager Jose Mourinho was irritated by his side's missed opportunities to win the match.
"When you don't play well and don't get the result, you are not upset. You are frustrated but you just accept it," said Mourinho.
"We missed 15 chances. The philosophy of both teams is important - one team wanted to win, the other wanted a point.
"But I don't blame Aston Villa. When they went a goal down they showed a reaction, and in the second half, they couldn't have done more."
Other Saturday results
In the early match Bolton went to second on the table with a 2-0 win at the Reebok Stadium over a hot-and-cold Liverpool side, with goals to midfielders Gary Speed and Ivan Campo.
Liverpool won 3-2 over Galatasaray in the UEFA Champions League midweek, but had no answer for Bolton who have not conceded a goal in their four matches at home so far this season.
At Goodison Park, Manchester City snatched a point late when City defender Micah Richards scored deep into injury time to cancel out Andy Johnson's first half goal for the home side, while in the late match new Middlesbrough manager Gareth Southgate's woes continued with a 2-1 loss away to Sheffield United.
Phil Jagielka scored a cracking goal in the 90th minute for the Blades to steal all three points after Rob Hulse and Aiyegbeni Yakubu had scored for Sheffield United and Middlesbrough respectively.