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.
Wesley started positively for Bremen as he tried his luck with a 25-yard effort which went wide of the goal in the third minute.
In the 10th minute Werder came forward again. Aaron Hunt's right-sided inswinging corner found Wesley and this time he kept his effort on target only to see Nikolay Mihailov make a decent save.
Twente's first opportunity came in the 15th minute when in-form midfielder Luuk de Jong fired on goal from 18 yards but Sebastien Mielitz was equal to it.
There was not too much in the way of goalmouth action, Marko Marin's cross - caught by Mihailov - a brief moment of interest in the 25th minute.
Again there were speculative efforts for the German side, first Hunt firing over from 35 yards then Hugo Almeida seeing his shot blocked by Rasmus Bengtsson.
Another Hunt corner saw Claudio Pizarro blast over the bar again and four minutes later, in the 35th, Marc Janko forged his first chance for Twente with a 25-yard shot which went left of the goal.
The game warmed up with half-time approaching. Marin's shot from the left but did not test Mihailov before his team-mate Almeida smashed a right-footed effort against the post.
At the other end Bryan Ruiz forced a brilliant parried save by Mielitz from close range.
Chadli hit one over for Twente and Wesley had an attempt blocked, while before the half was out Mielitz was again called upon to deny Ruiz, tipping the ball round the post from the Costa Rican's header.
A bright opening to the second half saw Denny Landzaat fire over for Twente and Chadli have another chance.
The action ebbed and flowed with more half-chances coming and going until the game burst into life in the 76th minute.
Frings, seemingly aggrieved with referee Alain Hamer's performance in general despite a lack of real controversy, was shown a straight red card for an outburst aimed at the official after he was adjudged to have fouled Ruiz.
Twente seized their opportunity and were ahead four minutes later.
After some decent build-up play, Chadli had yet another chance and this time he took it, drilling a low shot from just outside the box into the net.
Werder flailed as they attempted to respond and three minutes later it was very much game over.
The impressive Ruiz broke down the left and when his inch-perfect ball in found the head of De Jong, the midfielder gleefully headed home to wrap up three vital points.