NEC is bottom of the Eredivisie standings without a win from its opening seven games but it enjoyed the change of competition with an emphatic away victory over its fourth-tier opponent.
Another top-flight side among the goals was Cambuur Leeuwarden, which won 6-1 at Katwijk with six different players netting for the victor.
Heracles Almelo, which was a beaten finalist in 2012 when it lost 3-0 to PSV Eindhoven, is also through but it needed extra-time to see off fellow Eredivisie side RKC Waalwijk 1-0.
Midfielder Thomas Bruns struck the match-winning goal in the 116th minute.
In the other matches involving Eredivisie sides, ADO Den Haag and Go Ahead Eagles both claimed 3-0 wins, beating GVVV Veenendaal and DVS 33 respectively.