The game sprung to life midway through the second half courtesy of Adrian Lopez's superb individual goal for Atletico but Mame Diouf ensured a tense last 10 minutes when he drove home from inside the box for the hosts.
But it was the Spaniards who had the final word as the Colombian's 87th minute rocket set up an all-Spanish semi-final with Valencia.
With Hannover unbeaten at home and Atletico with just one loss in eight Europa league away games, it promised to be a tight match and so it proved in a niggly and lacklustre first half.
Diouf had an early sight of goal on 10 minutes, but fouled Atletico keeper Thibaut Courtois in his haste to connect and the man from Senegal dragged a shot across the goal a minute later when he should really have tested the Chelsea loanee from the edge of the box.
Sergio Pinto wasted one of a host of Hannover free-kicks by driving his shot against the Atletico wall from distance and the Germans again had a chance of a sight of goal when Diego was caught napping. Didier Ya Konan raced away from the surprised Brazilian only to slice the ball well wide of the target with options either side of him.
Atletico were content to play at their own pace and their best chance of the half came when a superb cross from Diego found Adrian in the box, who rose well but failed to steer the ball on target as the half remained deadlocked.
It was the visitors who started the second half the brightest. Diego's superb pass sent Falcao racing away, only for the Colombian to drag his shot across the goal with just Ron-Robert Zieler to beat.
But the visitors eventually made their pressure tell in the 63rd minute.
Adrian got on the other side of Emanuel Pogatetz and as Zieler committed himself, Adrian pulled the ball back, wrong-footing Christian Pander before slotting the ball home.
Hannover had failed to trouble Courtois all match but when Steve Cherundolo had a shot blocked by Joao Miranda with 11 minutes left, they found some new belief which paid dividends within seconds.
A long throw into the box caught out Diego Godin who could only head the ball into the path of Diouf; Diouf taking the ball on his chest before driving home.
Hannover boss Mirko Slomka played his last card by throwing Artur Sobiech into the fray, but the Pole had no chance to touch the ball before his side were down and out.
Diego found Falcao in space from the edge of the area and he smashed the ball past Zieler to send Atletico flying into the last four.