The Gunners move back into the top four although television replays showed van Persie looked suspiciously offside for the equaliser.
Boss Arsene Wenger said, "To come here and win is fantastic for the team," he told Sky Sports 1.
"It is much better but still not close enough," said Wenger.
"But you know we can gain belief - we have beaten Man Utd, we have beaten Chelsea away and I believe that it shows we have potential, we have character, fantastic character in this side but we have now to be humble and come back tomorrow and do our job."
Van Persie's first goal looked offside and asked if he cared, Wenger said: "I care because I do not want to score offside goals but I didn't see that frankly, I will check it again on the tape but (they) can sometimes go for you, sometimes against you."
He revealed that he gave his side a positive team talk at half-time. "I felt that we had chances, they had very little chances. I just told the team we want to go on and keep the pace and last longer than they do and come back very strong and show that we keep going. We had the quality and in the second half I feel that we were on top.
"We were 1-0 down and still won the game. There is no better test of character."