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.
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.
