Ajax eased into the fourth round of the KNVB Beker but fellow Eredivisie sides ADO Den Haag and NAC Breda were knocked out on Tuesday night.
Ajax had no trouble in brushing aside Hoofdklasse side ASWH 4-1.
Michael Van Dommelen scored for the visitors in the 29th minute to cancel out Viktor Fischer's early opener, but Ajax responded with two quick goals from Danny Hoesen and Siem De Jong to take a two-goal lead in at half-time.
Substitute Kolbeinn Sigthorsson then added a fourth with 19 minutes to go.
ADO Den Haag took their leave from the competition as they suffered a 2-1 defeat away to Eerste Divisie side MVV.
Rick Geenen scored in the 38th and 43rd minutes to put the hosts in control and although Mike van Duinen got one back in the 65th minute, they could not find an equaliser.
NAC's exit came on penalties at the hands of fellow Eredivise side Cambuur Leeuwarden.
NAC came from two goals down to force extra time, Anouar Hadouir scoring the equaliser in the 84th minute to add to a Mats Seuntjens effort in response to first-half strikes from Marcel Ritzmaier and Alexander Christovao, but Cambuur won the shoot-out 5-4.
FC Utrecht needed extra-time to win 2-1 away to Kozakken Boys of the Topklasse.
They fell behind to a 28th-minute opener from Arif Irilmazbilek before Jacob Mulenga replied in the 63rd minute and Jens Toornstra struck what proved to be the winner three minutes into extra time.
Michael Higdon's eighth-minute goal was enough for NEC Nijmegen to win 1-0 away to Eindhoven.
The Topklasse's IJsselmeervogels twice came from behind to upset the Eerste Divisie's Emmen 3-2 in a match which saw both sides finish with 10 men.
Alexander Bannink put Emmen ahead after just six minutes only for Thomas Verheijdt to reply four minutes later.
Quenten Martinus restored their advantage in the 19th minute and that is how it stayed until the decisive passage began 10 minutes into the second half.
First Verheijdt levelled the game up for a second time and then, four minutes later, Martinus collected a second booking.
The visitors used the extra man to grab a 66th minute winner through Achmed Ahahaoui and, although they had substitute Nabil El Gourari sent off late on, they hung on for the win.
JVC Cuyk dispatched fellow Topklasse side EVV 3-0 with a first-half brace from Kevin van Veen and a third from Caifano Latupeirissa two minutes into the second half.