Ajax took control after a lively opening, with Anton Gaaei breaking the deadlock shortly after the hydration break.
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Marcos Leonardo then doubled the advantage from the penalty spot after Oliver Edvardsen was fouled inside the box.
Volendam pulled one back after an error from Ajax goalkeeper Joeri Heerkens, who failed to deal with a back-pass from Youri Baas.
However, Arokodare provided the perfect response, scoring Ajax’s third goal to seal victory and make an immediate impact for his new club.
The Nigerian striker’s goal will boost his confidence ahead of Ajax’s competitive campaign.
Ajax next face Shelbourne FC on Thursday in the UEFA Conference League third qualifying round at the Johan Cruyff ArenA.