Striker Van Wolfswinkel headed home his first goal after 80 minutes and followed that up with a second two minutes from time to take his tally for the season to 21 in all competitions.
The 28-year-old former Netherlands international also collected the first trophy of a career that has seen him turn out for Sporting Lisbon, Norwich City and Real Betis.
Vitesse had finished runners-up in the Dutch Cup on three occasions, the last in 1990, and their triumph means they have qualified for the Europa League group stage next season.
(Reporting by Nick Said; Editing by Ken Ferris)
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