The Dutchman held off competition from Wayne Rooney to land the honour, which was won by Scott Parker last year.
Manchester United’s returning veteran Paul Scholes came third with Fulham’s Clint Dempsey in fourth.
Van Persie was recognised by his fellow professionals at the weekend, and he has now capped off a memorable season by winning both honours.
The last player to win both major awards in one season was Rooney in 2010, while the last Arsenalman to do it was Thierry Henry in 2004.
Van Persie’s 34 goals in all competitions have helped propel the Gunners into the top four of the Premier League, leaving them on the brink of UEFA Champions League qualification for next season.
FWA chairman Steve Bates said: “Robin was the overwhelming choice of our members this season and deservedly so after a magnificent campaign for Arsenal.
“The quality of his goals, attacking movement and consistency in a turbulent season for the Gunners has been eye-catching.
“The considerable size of his majority in the votes cast by our members reflected the general view that he has been the season’s most outstanding individual performer.”