Van Persie nodded home Song's cross in the second half to break the deadlock before finishing from close range after Thomas Vermaelen's near post flick-on.
Arsenal recovered well after being on the back foot for most of the first half as Dortmund faded after the break. The reigning Bundesliga champions grabbed a consolation goal in injury time through Shinji Kagawa.
Of Van Persie, Song told Sky Sports: "He's in unbelievable form. He's a very good player and a pleasure to play with and he shows how good he is in every game.
"He is very important and we hope he doesn't get injured because when he does, it is very difficult for us. We just need to keep him in good form."
Song said Arsenal did well to bounce back from a first half in which the visitors dominated, adding: "We took our time to get going, it was not easy but in the end it was really good and we got the result.
"We wanted to do well today so we're really happy. We showed good character.''