Van Persie chipped the ball over the goalkeeper from an acute angle with his right foot in the 57th minute for Feyenoord's second goal of a comprehensive performance at De Kuip. Danish international Nicolai Jorgensen had opened the scoring shortly before the half hour mark.
It's a first major domestic honour in the Netherlands for the 34-year-old, who returned to his Dutch football in January after spells at Arsenal, Manchester United and Fenerbahce.
Jens Toornstra added the third goal deep in stoppage time at the end of the 100th Dutch Cup final, interrupted early by flares set off by the rival sets of fans and fireworks thrown onto the field.
Feyenoord won the trophy for a 13th time as AZ finished runners-up for a second successive year.