Michel Jansen's side looked set to lose further ground on the leaders when Steven Berghuis opened the scoring after 40 minutes.
However, Youness Mokhtar equalised in the 69th minute before Dusan Tadic fired home a long-range winner deep into injury time.
Twente's victory - their first in four games - moves the club up to second and within seven points of Ajax.
The Eredivisie's top two meet in Amsterdam in a fortnight's time and victory for the visitors would put the holders under further pressure.
Frank de Boer's side were held to a draw for the second week running as they drew a blank against NAC Breda at the Rat Verlegh Stadion.
The withdrawal of skipper Siem de Jong due to a hamstring injury made a disappointing day for Ajax worse as they failed to capitalise on Vitesse's 2-1 defeat to PSV on Saturday.
At the bottom of the Dutch top flight, NEC no longer prop up the table after twice coming from behind against Utrecht to draw 2-2 and move up a place to 17th.
Roda's 2-1 defeat to Heracles handed NEC the opportunity to move off the bottom, which they achieved thanks to equalising goals from Michael Higdon and Jakob Jantscher.
A winner would have seen them move a place higher after RKC Waalwijk suffered a 3-2 defeat to Cambuur on Friday.
Feyenoord bolstered their hopes of qualifying automatically for the UEFA Europa League as goals from Ruud Vormer and Lex Immers secured a 2-0 win over Heerenveen, who had Christian Kum sent off 17 minutes from time.
Go Ahead Eagles failed to build on their surprise victory over Twente as they were beaten 2-0 by Europa League hopefuls Zwolle.
