The visitors were made to work hard for all three points at the Abe Lenstra Stadion, but eventually came through courtesy of two goals in the last 17 minutes from Dusan Tadic and substitute Torgeir Borven.
Hakim Ziyech had earlier threatened to give the hosts the advantage, firing straight at goalkeeper Nick Marsman from long range twice in the first half.
But after the break Twente stepped up the pace decisively - Tadic's shot after 73 minutes looping in via a deflection, before the Serbian turned provider for Borven to slide home at the back post.
Meanwhile, PSV Eindhoven's inconsistent season continued as they were forced to come from behind to snatch a 2-1 victory over Cambuur.
The 21-time league champions were on the wrong end of a shock on Sunday, as they fell to an Evander Sno double against RKC Waalwijk.
They started poorly again on Wednesday, with Elvis Manu firing home to give Cambuur the lead after just two minutes.
Phillip Cocu's side responded, though, and they levelled when a free-kick into the area deceived everyone, apart from Jeffrey Bruma - who was on hand to head in at the back post.
Jurgen Locadia completed the turnaround after 61 minutes. PSV sit seventh in the Eredivisie standings, with 32 points from 22 matches, 12 adrift of leaders Ajax.
Elsewhere, 10-man NEC Nijmegen and NAC Breda shared a 1-1 draw, as Rydell Poepon's 17th-minute strike was cancelled out by Rens van Eijden six minutes later - with the hosts forced to hold on after Tobias Haitz was sent off late on for a second booking.
Go Ahead Eagles and Waalwijk also shared a point apiece from a 2-2 draw after a remarkable finish at De Adelaarshorst in which Xandro Schenk thought he had won it for Go Ahead with a 91st-miunute effort, only for Romeo Castelen to respond a few seconds later.
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