The home side were the better team in the first half and Joachim Andersen saw his shot pushed away by the feet of Jasper Cillessen before a follow-up shot was blocked.
Twente then took a deserved lead in the 17th minute when Jesus Manuel Corona charged down the right flank before stepping inside onto his left foot and bending a precise finish into the far corner.
Kolbeinn Sigthorsson was left to fight for scraps upfront by himself for the majority of the half, but he did forge a half-chance when he chested down a long ball and drove a low shot at goal that was easily held by Nick Marsman.
Ajax came out in the second half a better side and Lasse Schone was unfortunate to see his swerving free kick glance off the top of the crossbar.
With 20 minutes remaining, Sigthorsson raced away from the Twente defence and sent a cool finish into the back of the net to earn his side a point.