The striker, in his first season with the new Premier League champions, beat midfielder Michael Carrick and goalkeeper David De Gea into second and third place, respectively.
Van Persie said: "This whole year has been incredible. I have to thank all the boys, the staff and the fans, who since day one, have been unbelievable. It's been an honour.
"It's made such a big impact on me and I'm still thrilled about what's happened in the last 10 months or so. Hopefully, we can do even better, crack on and win even more trophies."
Van Persie was also keen to pay tribute to Sir Alex Ferguson, who takes charge of his final game as United manager against West Brom on Sunday.
"It's only been a year I've worked with this great manager and great man," he added.
"But, honestly, I must say it's been a real honour and it's had such a big impact. Also in the way he is as a person, we get to know each other. Of course, I see him every day and the way he looks after his boys is like we're his own kids.
"He's always protecting us and looking after us. The meetings he does are incredible. It's been a joy and I'm proud of that."
The Dutchman, whose 25 goals drove United to the Premier League title, took his second award of the evening when his stunning volley against Aston Villa was voted United's Goal of the Season and Michael Carrick was named the Players' Player of the Year.