The Bundesliga side led 3-1 heading into the second leg of their last-eight contest, but were facing up to the prospect of extra-time as they entered the interval of the return meeting.
Swiss star Gokhan Inler offered Udinese a lifeline when he fired home from distance after just 15 minutes at Stadion Friuli.
Bremen’s Brazilian playmaker Diego then appeared to have put the Germans back in the driving seat after 28 minutes as he danced his way into the box before slotting home.
However, Fabio Quagliarella put the hosts back in front on the night as he found space to lift a delicate effort over the onrushing Tim Weise.
The Italy international was then on target again eight minutes later as he levelled the tie on aggregate with a composed finish.
Bremen refused to buckle, though, and on the hour-mark Diego put them in control of the contest.
With Udinese forced to chase the game it then came as no surprise when Claudio Pizarro put the seal on Werder’s progression after 74 minutes.
Elsewhere, Dynamo Kiev benefited from two goalkeeping gaffes by Mickael Landreau to ease past Paris St Germain 3-0 (3-0 on aggregate).
The tie was well-poised after last week’s goalless first leg at the Parc des Princes but PSG were no match for Dynamo seven days on with an early goal from Ismael Bangoura before Landreau punched the ball into his net.
Worse was to come for the 11-times-capped France custodian, who spilled a free-kick to allow Ognjen Vukojevic to slot home Dynamo’s third just after the hour mark.
Shakhtar Donetsk, meanwhile, eased their way into the last four with a 2-1 win against Marseille (4-1 agg).
Holding a 2-0 lead heading into the second leg the Ukrainian outfit never looked like allowing that advantage to slip.
They were forced to soak up some early pressure at Stade Velodrome but went ahead just before the half-hour mark through Fernandinho.
The Brazilian was able to squeeze home from the tightest of angles and virtually end the game as a contest.
To Marseille’s credit, they refused to accept their fate and snatched an equaliser two minutes before the interval.
France international Hatim Ben Arfa was the man to force the breakthrough as he rounded off a flowing move by lashing into the back of the net.
The hosts did all they could after the break to find a route back into the game, but it was to no avail.
Then, with time running out, Luiz Adriano put them out of their misery as he snatched a late winner for Shakhtar.