Gregory van der Wiel scored the winner 12 minutes from time to leave the Amsterdammers six points clear at the top with two games to go.
Ajax also have a 24-goal advantage over closest rival Feyenoord, so not even two defeats should deny it the league trophy.
Theo Janssen's penalty put Ajax ahead in the 30th minute only for Leroy Fer to level after the break.
Van der Wiel then intervened to end Twente's own slim title hopes.
Otman Bakkal scored the only goal to keep Feyenoord's faint hopes alive and knock opponents AZ out of the race for the championship.
Bakkal struck early in the second half as AZ slipped nine points behind the leader and will now focus on ensuring a top-three finish and European football next season.
PSV Eindhoven maintained its hopes of a top-three finish with a 3-1 victory at 10-man Roda JC.
Goals from Dries Mertens, Marcelo and Memphis Depay secured the points for the visitor, which also missed a penalty through Jeremain Lens in an incident which saw Roda goalkeeper Pawel Kieszek sent off.
The result enabled PSV to record its fourth successive win as it mounts a late bid for an automatic European spot.
Sanharib Malki scored the only reply for Roda, whose challenge for a Europa League playoff place was ended with the defeat.