With the game finely poised at 0-0 early in the second half Scottish referee Craig Thompson awarded a penalty to the Dutch side as striker Luuk de Jong went down outside the box with Joel Matip in pursuit.
Replays confirmed Matip - who was sent off - had not touched De Jong, who seemed to clip his own heels. De Jong scored the penalty to give Steve McClaren's side a slight lead to take to Germany.
Schalke's Atsuto Uchida carried the fight early on with a 20-yard shot in the seventh minute which Twente goalkeeper Nikolay Mihailov saved easily.
Tim Cornelisse forced a chance at the other end as he crossed for De Jong, whose headed was caught by Timo Hildebrand.
Twente had another opportunity as Roberto Rosales sent in a centre which Nacer Chadli headed on target without troubling Hildebrand. Schalke had a chance of their own as Lewis Holtby's 25-yard sighter flew over the bar.
Schalke midfielder Jermaine Jones' shot sailed over while Peter Wisgerhof had a chance for Twente in the 31st minute, shooting from the right only for Hildebrand to save.
Ola John was lively at times and crossed for Chadli to shoot, though the left-footed effort from close range was woefully off target.
Schalke's Ciprian Marica fired wide from outside the box as half-time arrived.
The deadlock was broken shortly before the hour mark in controversial circumstances.
John played a smart low pass towards the right which freed De Jong to charge towards goal. He was pursued by Matip but when the striker clipped his own heel outside the box, referee Thomson and his numerous assistants got it completely wrong by awarding a penalty and sending Matip off.
De Jong gleefully smashed the resulting spot-kick into the top left-hand corner to put his side ahead in the 61st minute.
Schalke's response was hardly emphatic as De Jong's poor header wasted a chance to make it 2-0 nine minutes from time.
Jones headed over from Julian Draxler's corner while Draxler himself fired a left-footed effort over the bar.
Twente continued their search for a second right up until the final whistle as the likes of Chadli and Emir Bajrami whipped in crosses which were not capitalised on.