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The game was a succession of half-chances until the 74th-minute dismissal of Torsten Frings swung the balance in the Dutch side's favour. Twente seized their chance, Nacer Chadli quickly finding the breakthrough before Luuk de Jong confirmed the win just moments later.
Rangers suffered their first defeat in the Champions League this season as Roberto Soldado grabbed a double to hand Valencia a 3-0 victory at the Mestalla.
He netted on both sides of the interval, before Tino Costa sealed the win late on, as the Spaniards leapfrogged Rangers into second place in Group C. Steven Naismith hit the woodwork twice on a frustrating night for Walter Smith's men.
Rubin Kazan and Panathinaikos saw their hopes of qualifying for the Champions League knock-out stage recede further with a goalless draw in Russia.
The hosts were the only side to carve out any chances of note, with Alexander Ryazantsev's fierce header which goalkeeper Alexandros Tzorvas did well to turn away the closest they came to breaking the deadlock.
Hapoel Tel Aviv recorded their first ever point in the Champions League with a 0-0 draw against Schalke - but Bloomfield was rarely lit up on a lacklustre European night.