Goals from Abou Diaby, Emmanuel Eboue and van Persie ensured the Gunners bounced back from their disappointing draw with Blackburn and moved within seven points of leaders Manchester United with a game in hand.
When asked if the Gunners could win their seven remaining games to clinch the title, van Persie told Sky Sports 1: "We needed this, we needed to get a win so we are pleased.
"We worked hard and deserved it. I believe we have it in us (to win all our remaining games). There are seven games to play and it us up to us.
"We have it in ourselves and we have to prove it. We are a great team and it is down to us now."
Keeper Jens Lehmann was delighted after making his 200th appearance for the Gunners, and his first since rejoining the club, after Manuel Almunia suffered an injury in the warm-up.
The German was maybe fortunate to see out the game after bringing down DJ Campbell in the box, but the ball ran for Gary Taylor-Fletcher to score.
He said: "Sometimes life is about surprises and you never know what is coming. We are back in it and lets see how it goes."
When asked if he thought he would see red for his foul, he replied: "I don't know, the rules sometimes changes. But I don't care, if that was my final game I am pleased as we won."
