The 20-year-old will join Arsenal "on a long-term contract for an undisclosed fee and subject to a medical", the London club said on their website.
"I'm delighted to be joining a great club like Arsenal, this is a dream come true for me," the under-21 international said.
Arsenal manager Arsene Wenger, whose team wrapped up their third English title in seven seasons with a 2-2 draw at Tottenham Hotspur on Sunday, said: "Robin is a great young talent and a fantastic signing for the club.
"He has a great left foot and is a great passer of the ball with excellent vision. He is a versatile player and is comfortable playing as a striker or on the wing."
Van Persie joined Feyenoord as a 14-year-old and over the last three seasons has scored 14 goals in 77 matches for the club. He played in the 2002 UEFA Cup final when Feyenoord beat Borussia Dortmund 3-2.
Arsenal first made an offer for the forward in the January transfer window but the two clubs failed to come to a agreement before it closed.
He is set to be officially registered on May 17, when the transfer window re-opens, and is expected to join the squad for pre-season training in July, Arsenal said.
Despite his six goals, Van Persie has had a difficult season at Feyenoord, falling out with coach Bert von Marwijk and spending much of his time on the bench.
His agreement to join Arsenal comes as Wenger overhauls their forward line, with the contracts of Dutchman Dennis Bergkamp, France's Sylvain Wiltord and Nigerian Nwankwo Kanu all expiring at the end of this season.
Arsenal signed 20-year-old Spanish forward Jose Antonio Reyes from Sevilla in January.
The two youngsters should complement 30-year-old Frenchman Robert Pires, still Wenger's first-choice on the left side of midfield, and main striker Thierry Henry.
"The squad is full of players at the top of their game," Van Persie said. "I'm looking forward to meeting all my new team mates in the summer and helping the club to challenge for more honours."
