Van Persie, the league's top scorer, took his tally for the season to 30 with two goals in the 72nd and 80th minutes but Steve Morison rescued a point for the visitors with five minutes to go.
Arsenal have 67 points with one game left to play this season. Fourth-placed Tottenham Hotspur, at Aston Villa, and fifth-placed Newcastle, at home to leaders Manchester City, play their games in hand on Sunday.
Manchester City and Manchester United are locked together on 83 points. United host Swansea City on Sunday.