The amateurs grabbed an early lead through Thijs Hendriks after he intercepted a pass from PSV defender Mathias Jorgensen.
Jurgen Locadia equalised for the holders from close range after 25 minutes before Jorgensen and Timothy Derijck scored following corners to give visiting PSV a 3-1 advantage at half-time.
Ivo Rigter pulled a goal back for Achilles in the 63rd minute and then Georginio Wijnaldum missed a penalty for PSV.
Elsewhere, Swede Viktor Elm notched a first half hat-trick to lead AZ Alkmaar to a 4-1 victory over second tier Veendam.
Jozy Altidore converted a penalty after 76 minutes before Tom Overtoom grabbed a consolation goal for visiting Veendam.
On Wednesday, there were wins for big guns Ajax Amsterdam, Feyenoord and Twente Enschede.
- Timothy Derijck
- Georginio Wijnaldum