The home team failed to extend the lead and Twente substitute Adnane Tighadouini equalised after a mistake by Joel Veltman. Lasse Schone handed Ajax the win by converting a penalty in the 71st minute.
Five minutes from half-time, Steven Bergwijn equalised with a curled shot.
PSV started the second half without injured top scorer Hirving Lozano, but Bergwijn secured the victory with another fine shot and his second goal.
Also on Saturday, AZ Alkmaar had difficulties with VVV-Venlo in Alkmaar. The home team was not able to break the defence of the visitors and the match ended 0-0.
PSV now lead the Dutch table with 61 points after 23 matches, followed by Ajax on 54 points. AZ are third with a total of 47 points.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)