Tim Cahill put the Toffees ahead in the 61st minute, heading home his 100th career goal.
But Van Persie had the last word to keep Arsenal in touch with the Champions League placings.
Gunners boss Arsene Wenger was pleased with the point.
"When you are 1-0 down with a minute to go and you get a point you are relieved. It was a game where two teams gave everything and we showed quality and character.
"We showed fighting spirit, passion and intensity and Robin van Persie scored an exceptional goal.
"We can pass the ball better but you have to give a lot of credit to Everton. They play in an intelligent way and play a kind of game that made a passing game very difficult.
"It is a game we wanted to win but it was one we couldn't afford to lose so maybe it is not too bad."
David Moyes the Everton boss was disappointed not to have claimed all three points.
"We feel bad because the players were terrific. We deserved the victory and to have it taken away right at the end is very disappointing.
"Tim Cahill is doing a magic job for us. He has got a goalscoring record like a centre-forward. He is scoring like a striker should. I wouldn't say he is making it look easy but he is doing great.
"He went off injured but I don't think it is too bad. He took a kick on the foot but hopefully it is not too bad."