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.
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.
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.
