Central defender Douglas (pictured) notched the opener 10 minutes after the break before Verhoek scored his first goal for Steve McClaren's Twente seven minutes from time to seal the victory.
Lacklustre PSV Eindhoven were still good enough to beat strugglers VVV Venlo 2-0 with Tim Matavz and Memphis Depay on target while Ola Toivonen missed a penalty after 70 minutes.
PSV are fourth with 54 points, ahead of Feyenoord and Heerenveen on goal difference after the pair registered 3-1 wins at home to NAC Breda and at Groningen respectively.
Feyenoord's Otman Bakkal struck from distance after three minutes to put his side ahead but Omer Bayram equalised with a long drive that surprised goalkeeper Erwin Mulder.
However, Feyenoord restored their lead through Ruben Schaken with Sekou Cisse also on target before the interval.
Heerenveen's Bas Dost took his tally to 25 as the league's leading scorer notched a double in the win over Groningen.
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