However, since then only an FA Cup triumph has followed as Arsenal – who did reach the 2006 Champions League final – have found themselves overtaken by Chelsea and Manchester United.
This season, both Barclays Premier League leaders Liverpool and Aston Villa have also moved ahead in the race for the title.
However, Van Persie, 25, insists that under the guidance of Wenger, who has been at the helm for more than a decade, this side can go on to achieve greatness.
"I feel the same as the fans: I want it as badly as them and it can be frustrating when it doesn't happen," said Van Persie, who has hit five goals in his last seven games.
"I am a positive person though, and I think positively, and I think the fans should do that as well. We did not win everything in the past couple of years, but we are doing absolutely everything we can to change it."
Van Persie added: "The main thing in 2009 is to finally get silverware, but there is a clear picture here of how to achieve that.
"We need to stick to it –because this is how Arsenal is. The Arsenal way is in our system now, it grows into you, and I don't think it's clever to change your philosophy.
"Things take time, and you often see clubs sacking managers for fun, and they don't get anywhere. You need a long-term strategy.
"At Arsenal, the players have time to grow into the system, and if we stick to our principles we will get silverware."
Arsenal are undefeated in the league for seven games. However, with captain Cesc Fabregas and winger Theo Walcott both injured, Wenger admits he would like to bring in at least one fresh face.
The most likely candidate continues to be Russian playmaker Andrei Arshavin, with an improved bid of £12 million said to be on offer to Zenit St Petersburg for the unsettled 27-year-old.