Nikos Machlas grabbed a first half brace, while Rafael van der Vaart and Cristian Chivu (pictured), the latter from a penalty, also got their names on the scoresheet in the first period.
After the break, Tomas Galasek fired home Ajax`s fifth, while Jan Vennegoor of Hesselink grabbed a consolation for the visitors after 72 minutes.
The win keeps the Amsterdam outfit in fourth place, with 33 points, while Twente stay in 12th place on 20 points.
