Heracles led 2-0 after 30 minutes but home side Achilles came storming back with a goal either side of the break before three more after the hour mark secured a famous win.
Holders Ajax Amsterdam grabbed three second-half goals to break the stubborn resistance of second division Veendam.
Mido headed the opener after 61 minutes before Luis Suarez and Christian Eriksen sealed a last 16 spot.
Groningen needed penalties to get past ADO Den Haag after their clash ended in a 1-1 draw with Lex Immers putting ADO ahead before Jonas Ivens equalised seven minutes from time.
PSV Eindhoven and Twente Enschede also progressed after beating Spakenburg and Zwolle respectively earlier this week.
