It is Vitesse's first major trophy in their 125-year history.
Striker Van Wolfswinkel, 28, headed home his first after 80 minutes and followed it up with a second two minutes from time.
Vitesse, who sit sixth in the Dutch League, qualify for the Europa League group stage next season.
Their cup final team included three players on loan from Chelsea - Brazilian midfielder Nathan, 21, United States defender Matt Miazga, 21, and 22-year-old midfielder Lewis Baker.
AZ Alkmaar's team included goalkeeper Tim Krul, 29, who is on loan from Newcastle United.
Former Norwich City striker Ricky van Wolfswinkel scored two late goals as Vitesse Arnhem beat AZ Alkmaar 2-0 to win the Dutch Cup final in Rotterdam.